I've always liked Trig and I went the Trig route: Sin (45) = .7071 Sin x = opp/hyp .7071 = opp/25, opp=25(.7071) opp = 17.67*2 (for the full side of the square) 35.35 * 35.35 (area of a square) = 1249.811 a little more precise than just 1250, which if you're cutting wood, the .19 is about the width of a saw blade(thin kerf) and can make a difference
Which is why you should do it that way. Less precise because you didn’t use enough decimal places. If you use more decimal places the value will be 1250 exactly.
Do not complain if you don't like this teacher. He is helping people that are learning math, not people that already can math. Keep on teach, teacher 👍
My issues with this teacher are a) the length of time he takes to explain things, and b) that he does not always demonstrate the easiest solution. If someone is learning the basics of mathematics, the simpler, or more interesting, you can make it seem, the more likely they are to want to learn more. And the benchmark of a good teacher is having a student WANT to learn more, not leaving them feeling they need more help, or that it is too complicated to bother with.
@@davek6415, first, a student who is not already familiar with this process or why it works may not absorb it as fast as someone who already understands similar math. He isn't trying to teach those who already grasp it. Second, perhaps he feels that his method is the one that seems to make the most sense, is the most intuitive to a student new to such problems. That is not always the same thing as the easiest process, the fewest steps to obtain the correct answer. A good teacher is someone who understands the psychological processes of their students.
@@johnwest7993 Well, my son, a math teacher agrees with me that the easiest solution is the best solution. Why take the students around the world to solve this ? Imagine taking such a long method on a timed test. You could already be on the fourth problem by the time you work this one out using his method.
You are at the centre of a square, 25' from a corner, so, 25' from each corners. Take any two adjacent corners, and you have a right triangle, with one face of the square as it hypotenuse. So, take 25^2 * 2 to get the square of length of the hypotenuse. So, 25^2 * 2 = 625 * 2 = 1250. You don't need to find the square root of 1250 just to square it. Meaning that you already have the area of the square room : 1250 square feet.
Cool approach. Thanks. I did it in my head by just knowing that the diagonal of a square is √2 times the length of a side. So 50/√2 is the length of a side. Square that: (50/√2)² = 2500/2 = 1250.
Triangle, area of = base times height divided by 2. You are 25 feet from the 3 corners of a right triangle therefore the base is 50 feet and the height is 25. You have 2 of those triangles therefore the 2’s cancel, so 25 times 50 is 1250. A second way is to recognize that a square is comprised of four right triangles that form two smaller squares that have a side length of the distance from the center of the larger square to one of its corners. 25*25= 625 is the area of each small square times 2 = 1250 sq ft. A) youngster, I enlisted in 1971 and retired in 1995 from the Navy. B) I fold the flag when we bury one of Uncle Sam’s Misguided Children (For those of you that are confused, there is branch rivalry, however we band together when an outsider gets involved.)
It took me less than 20 seconds to read your explanation. I should have gone straight to your comment. Sadly, I watched John’s 18 minute lecture first. It was, how shall we say, an ordeal.
Easier mental math - from the center to the corner is 25 and also to all corners. So, From the center to two corners forms an isosceles right triangle which is 1/4 of the square. Base times height is 25*25 or 625. The square is twice that value (or (625/2) * 4), or 1250.
@@walkfaster Ok, here's another way. Slide rule enthusiasts should not be left out of the fun. If you're 25 feet from the corner, the area of the square is 10^(log(25)*2+log(2)) = 1250. log(25)*2 is the log of 25^2. log(25^2)+log(2) is the log of 25^2 times two. You raise ten to that power because the base ten log of a number is the exponent you need to use against a base of 10 to get the number. Note that any log base will do, so e^(ln(25)*2+ln(2)) also equals 1250. However, log of 25 is hard and any log will do, right? How about 25^(2+log(2)/log(25))? That also equals 1250, but is trickier. 2 is log base 25 of twenty five times 2, because log base 25 of 25 is 1. 1*2, Terrance Howard notwithstanding, is 2. log(2)/log(25) is the base 25 log of 2. That converts the base ten log of two to its base 25 counterpart. Addling logs is like multiplying the bases. Adding it to the log_25(25)*2 is, again, multiplying by 2. You get the base 25 log of 25^2 * 2. Take 25 to that power and you get 1250. Now, to my calculator to see if that worked. I'm pretty sure it did.
@@darkdelta A EE professor I had was catching grief from a computer major who thought computers would soon obsolete engineering. “Young lady,” he said, “we put men on the moon and returned them safely with slide rules and three digits of precision.” I will never forget that comeback. That was Professor Dougal at UT Austin. An amazing guy.
@@johnnyragadoo2414 Shut herdown pretty ole quick. I can remember talking with some colleagues, again in the 1960s, how one day we'll get electronic adding machines. We never considered multiplication and division. Then Burroughs came out with a desktop calculator, it sported a Nixie tube display, among other features, Now my phone has more computing power than some mainframes back then,
At age 76, I worked this out in my head. The 4 isosceles triangles form two squares with side 25. So 2 x 25 x 25 = 50 x 25 = 100 x (25/2) = 100 x 12.5 = 1250
Construct your mental box at 45 degrees to your illustration In short 50X50 simple 2500 square feet, half of that is 1250. Yes I did Pythagorean Theorem as well to confirm,, and I did the hypotenuse of a 45 degree triangle is A X 1.414. 50' divided by 1.414 is about 35 (plus a little) and 35X35 (plus a little) is about 1250 Three methods. The 50X50 divided by 2 is by far the fastest easiest.
Length of a diagonal of a square= a✓2 (where a is the side of the square) As such , a✓2 = 50 a = 50/✓2 ft Area of the square= a^2 = (50 ft/✓2)^2 1250^ft^2/2= 1250 ft^2
OMG! Between the poster's long-winded, repetitious verbalizing and the commenters, I have to say that I now understand why America is 37th in the world in education. I was a not a good math student in school (now 75!) but I had this figured out almost in my head in a few minutes. Too much talk and too much complicated explanations that didn't explain the basic "need to know", i.e. a sq + b sq=c sq, and c=25, sq = 625. a and b are equal, since the room is square, and a sq is half, or 312.5, and b sq is half, the sq root of which is 17.6776, so the area of one of 8 triangles (base times height) is 17.6776 squared=312.5/2 (or a + b)=156.250 x 8 (triangles) =1,250. Some of the explanations in the comments make no sense, and by the time the poster was done yapping about incidentals, I had already stopped listening. That is the problem in school! It was the problem 60 years ago when I was in jr. high. God help our children!
Almost in your head....means you used caculator. I would have needed a calculator too. Some of these commentors are suggesting different ways to find the solution so they are adding value. Had I not skipped to comments I would have used method described on video.
It's a 45/90 triangle, so one leg is hypotenuse/sqrt(2) = 50/sqrt(2). Don't have to calculate what that is. Area is side^2, so 2500/(sqrt(2))^2 or 2500/2=1250. No calculator.
@@PaulTaylor-Young To be fair to the guy who posted the video, in a math test they probably want to know that you know all of it. In the real world there are definitely easier and quicker ways to do it, but all rely on you knowing the basics and being able to visualise it.
Draw the diagonal of the large square. The diagonal will be twice the distance from you to the corner, so 50 feet. This divides the square into 2 triangles. Choose either triangle and construct a square on each side of it. The square on the hypotenuse is 50 x 50 = 2500 square feet. By the Pythagorean theorem we know that the sum of the other two squares is also 2500. Further, the sides of the smaller squares will be equal, so their areas are also equal, and will be 2500 / 2 = 1250 square feet. Now notice that the original square had side length exactly the same as the sides of the two smaller squares. Therefore the area of the original square is exactly the same at 1250 square feet.
@@raamannair8072 Um, just before that it says you are in the *center* of a *square* room. Since this is a geometry problem, it is reasonable to interpret “square” as geometrically square, and “center” as the literal geometric center, so the corners are equidistant from you. If this were a geometry test you’d get zero marks for not demonstrating that knowledge. If I were marking it you might get -1 for being pedantic and actively denying this geometric truth.
Without watching the video the easiest way I found was to create a triangle using two sides going to corners with a side of the square being the hypotenuse. So the length of the side of the square is x^2 = 25^2 + 25^2. This becomes x2 = 1250. You can really stop there are the area of a square id L x B. And, in this case X^2 = L x B = 1250.
Like a lot of people out there I’m old and retired. Been too many years to remember this stuff. I ‘m happy for a chance to study these again. Love your explanations. Hopefully it will help keep my mind sharp
We are so impatient that we tend to forget that we are not the only one in the room. I already know most of what is discussed on this Channel, but I realize that he is catering for the whole class. It's the mark of a good teacher. He makes it easier for the slower one. Let's therefore remember those whose struggle with these concepts.
As a teacher myself, I have learned that other people (students) explaining their methodology can ‘click’ for some students who don’t get the teacher’s explanation. ** Even if they sometimes use the exact same words. 🤪
It's also 2 right triangles with base of 50ft and height of 25 ft. 2 • 1/2 • b • h (but 2•1/2 = 1) b • h = 50 × 25 = 1250 ft^2 Only one calculation necessary!
I looked at it like four triangles. If you put two of the triangles together you have a square with sides 25 x 25. 25 x 25 = 625 Sq ft 625 x 2 = 1250 Sq ft
If a person doesn’t know the method of figuring out the area of a triangle, there’s perhaps an easier way to solve this: if you start at a corner and walk to the middle of the room, you’ve gone 25 feet; if, instead of continuing on to the _far_ corner, you turn right and go to the _next_ corner, you’ve gone another 25 feet; if *now* you imagine that you’ve just walked down two adjacent sides of an imaginary square that’s half in the room and half outside the room, then that square is 25 by 25, only half of which is in the room; 25 by 25 equals 625; you can divide that in half to get the area of the imaginary square that’s in the room, *but* , since the area of the imaginary square that’s _in_ the room takes up only one quarter of the room, we can imagine a second imaginary square of the same size that takes up another quarter of the room - in other words, don’t divide the 625 in two - - instead, we see that 625 square feet is the area of exactly *_half_* the room; so, simply double 625 to 1,250 and you now have the total area of the room.
This method and explanation started out good, but "imaginary squares" weren't needed. Just calculate the area of that right triangle with base and height 25. Multiply by four. Or multiply by four first so you don't need to calculate 625/2.
The square room can be thought of as 4 right triangles each with a base = height = 25. Area of each triangle is 1/2 × 25 × 25 Since there are 4 such triangles. Total Area of Room is 4 × 1/2 × 25 × 25 = 2 × 625 = 1250 FT² So for any Square Room with the distance given from corner to center or vice-versa the area of the square is that distance given ^2 (squared) and then doubled. So if you are in the center of a square room 10 FT from the corner. The area of the room is 2 × 10^2 = 2 × 100 = 200 FT²
Or, the really easy way to solve it; 2 x 25 x 25 as you're in the centre of a square and 25 from the corner, that means that you can divide the room into 4 diagonally and make 2 squares of side length 25. Thus if each square has a side of 25 then twice the square of 25 is your answer. Which is 1250 ft squared.
For a unit square, the sides are 1 inch, and the diagonal is the square root of 2. All squares will have this same ratio. Therefore, if a square has a diagonal of 50, its sides are 50 divided by the square root of 2. The square of that is the area.
Form your diagram, Area of a triangle = 1/2 * height (h). Base =50, h = 25, but area of a square = 2* area of triangle. So area = 2. (1/2.50). 25= 1250 sq feet. OR. From your diagram, x= side of square. So, by Pythagorus, x(sq) = 25(sq) + 25(sq). But area of the square = x(sq). So area of square = 2(25(sq)) = 2(625) = 1250 sq feet. I appreciate these 2 solutions are linked as they both give 2(25.25) = 1250 sq feet..
The hypotenuse is 50 feet as you are in the center of a square room, this also means that the length = width (a = b) a^2 +b^2 = 50^2 a^2 + a^2 = 2500 2a^2 = 2500 2a^2/2 = 2500/2 a^2 = 1250 (sqr rt) a^2 = (sqr rt)1250 a = (sqr rt) 1250 a & b = 35.36 ft (rounded to the nearest 100th) a*b = 1250 ft^2
Just because it's a square, it doesn't mean you need to figure out square roots. If you are 25 feet away from a corner, and you are in the center, then you can divide the square into two or four triangles. The area of a triangle is 1/2 the base times the height. 25 is the base and the height for a quarter triangle. 25 times 25 is 625. that is two quarter triangles, which is half of the square. 625 plus 625 is 1250. You can also do two triangles, which is easier. 25 + 25 is 50. 50 is the base of a half triangle. 25 times 50 is also 1250. You don't need to halve the answer, because there are two triangles, when you bisect the square from corner to corner.
I just turned the room into two equal triangles. .5 × base × height is all you need for the area of a triangle. So. .5 × 50 ×25 = 625 then double that to 1250 since two triangles make the square.
Er, if you are in the centre then you are a distance “a” from each side and the length of each side is 2a so the area is 4a^2. Using Pythagoras, applied to the triangle formed from the 25ft diagonal from you to the corner and the distance “a” from you to each side shows 2a^2=25^2=625. So if the area is 4a^2 then the area is 625 x 2 = 1250ft sq.
I did it by finding the area of an interior right triangle with side lengths of 25, then multiplying by 4, because you have 4 of those interior right triangles and the combined area of those 4 interior triangles equals the area of the square room. Thus, the area of one of the triangles = 1/2 * (25*25) = 1/2 * (625) = 312.5, so then 312.5 * 4 = 1,250 sq. ft. for the area of the square room.
What a long winded way of doing it! Just form the right angle in the centre, by going to an adjacent corner, not the opposite one, so you have a triangle one quarter of the square’s area. The area of this is simple: area of a triangle is half base x height, in this case 12.5x25. But as there are 4 such triangles, the area of the square is therefore 12.5x25x4, or 12.5x100, ie 1250. No need for Pythagoras’ theory etc
I used Pythagoras and area of square formulas but made the length of one wall the hypotenuse. Center to each corner is 25. So 25^2+25^2=625+625 so one wall is 1250. Square root if 1250 is 35.355 for one wall. Area = LxW or 35.355x35.355 or 1250.
To the corner there are 25 feet. That's half the diagonal. We can use that diagonal in the formula. (50ft)^2/2=1250 ft^2 (the whole process was to do 25, *2, ^2, /2, = .) If we don't remember that formula we can easily get there with Pythagoras: 50^2=side^2 +side^2 2500=2*side^2 1250=side^2 Another option is to think that those 25ft semi-diagonals trace 4 right triangles inside the square. 25ft*25ft/2=312.5ft^2 *4 =1250ft^2
25ft to the centre, so diagonally corner to corner is 50ft. Two sides of the room and that diagonal form a right angle triangle where the other two angles are 45 degrees. The hypotenuse of such a triangle is √2 times the length of each of the other sides (you can check that with Mr Pythagoras) so the side length of the square room is 50/√2. Therefore the area = (50/√2)² = 2500/2 = 1250 ft²
That's exactly how I did it. This is the second video of John's that I have seen where he did not use the fact that the diagonal of a square is √2 times the length of a side. I think he just wants to show the Pythagorean steps, which is just fine.
Two sides of a triangle, equal to each other, with the hypotenuse being third side. Each side is25. So 25 squared plus 25 squared equals the hypotenuse squared, which is the area of the square.
Pythagoreas. a^2+b^2=c^2. In this case a=b for the sides of the room. The diagonal of the square is c, which is double of 25 = 50. c^2=2a^2 = 2(area) 50*50/2 = area. 2500/2 = 1250. Did it all in my head.
50^2 = 2X^2 (Pythagoras) => 2X^2 = 2500 => X^2 = 1250. The area = X x X = X^2 = 1250 (feet^2, in this case.) No need to solve for X here, given the fact that it is a square room.
@@Kleermaker1000You actually need to solve for X. Because you're in a square, and your distance is from the corner, not the middle of an edge. Need to multiply your 25' by sin (45) = 41.545 Square that, gets you 1810.078 sf EDIT - Aw hell. My calculator was on radians for some reason. Changed to degrees and wouldn't you know it, 1,250sf
The answer is firstly, one has to figure the length of one of the sides of the square. One knows that one is in the middle of the square where one is standing 25 units from a corner. Now, this is actually 50 units from corner to corner and that this length is 45 degrees at a corner. So, one uses the formula cos theta = x/h , base/hypoteneuse. So, x as one of the legs of the square is x = cos(45 degrees) * 50 and this gives you the leg of the square. Area of the square then becomes ((cos45 degrees)*50)^2.
I made a small right triangle in my head that had a hypotenuse of 25 ft. I ended up with sqrt(625/2) ft. for each triangle side. Therefore, multiplying this by 2 to get the length of each side of the big square (same as multiplying everything inside the sqrt by 4), you get sqrt(625 × 4 / 2) or sqrt(625 × 2) or sqrt(1250), and, because the area as this times itself, that just undoes the sqrt and the area is 1250 sq. ft. But, yes, your way was a bit quicker.
A rather easy solution to this problem is if you’re in the center of a room and 25 feet from the corner and the area of a triangle is 1/2 the length times the height then 25×25=625 Now you divide that in half 625÷2 =312.5 And now you have the area of a triangle, that is equivalent to 1/4 the area of the total of the square which means multiply times four and you get the area. 312.5x4=1250 Of course, if you’re thinking ahead of the game then 625 would equal half the area of the room because it’s two of your triangles that would make up the total of the area of four triangles so you could just multiply that by two to get the same answer. Sounds more complicated than it really is. But what I’m basically saying Is calculate the area of a triangle where a side of the square is the hypotenuse of a right triangle where the two sides are 25 feet which is equivalent to 1/4 of the total area of the square.
Except stop after 25x25=625 because you have the area of the triangle that is one half of the square. (Corner 1 to corner 2 to corner 3.) Just multiply that by 2.
It is a right, equilateral triangle and an hypotenuse of 50. A 1, 1 similar triangle has a hypotenuse of square root(2), or remember this relationship from prior interactions with this triangle. If we chose a similar triangle with a hypotenuse of 50, the sides are the 50 divided by the square root of 2. The square’s area is one side squared or 50 squared divided by the square root of 2 squared or 2500/2. So area is 1250 square units.
If you're in the middle of the room then the hypotenuse is the length of 25'' therefore A squared + B squared = C squared or 625'. A = B so A squared X 2 = 625'. The room is approximately 17' 8" X 17' 8" (17.6776695). 17.8 X 2 = 619.52
@16:40 Keep in mind that X is only approximately equal to 35.35 ft but it is exactly equal to the square root of one half the square of the hypotenuse. The area of a SQUARE is equal to one half of the square of the hypotenuse and will equal the square of one side.
Square of the hypotenuse divided by 2. The lesson to take away is that you don't always need to know the side length to work out an exact area of a square.
The diagonal is 50 but we want a side length to find the square's area, so draw another square around this one (in your head of course) turned 45 degrees so that the corners of our square are in the middle of the sides of the new one. We have the side length for the new square as a given (near enough) 50'. So the area of that square is 2500 sq ft, and it is twice the size of our original square so our square is 1250 sq ft. Takes much longer to write than it does to work out. Easy mental arithmetic.
Area of one triangle here with 50' hypoyhenus and the height from the hypothenus to the corner, is 25. Area of the triangle is then 625, and the area of the full square is 1250.
A simpler way. Knowing that from center to two adjacent corners of a square forms a right triangle with each line equidistant and has the side S of the square as the hypotenuse. Let's call this line C which is given as 25ft. So C^2 + C^2 = S^2. 25^2 + 25^2 = S^2 = 625 + 625 = 1250sqft. Since the area of the room is S^2 we already solved our problem.
I knew that hypotenuse of 45 degree triangle was 1.41412 so I divided it into 50 giving 35.357. Then multiplying the side squared = 1250.141. Area. If it had been a 60/30 triangle or rectangle with relationship of hypotenuse 1.732 into what ever number was the center of the rectangle. Your way was my first as the numbers were arrived without calculating the length of sides. However in any test I ever took 2 to 3 components were derived from initial answer to first part, resulting in being critical to calculate the correct answer for part one of the details reflecting each unknown measurement of the object. In this case, what if the square became a cube and volume requested. Excellent find, I will pass this on! At 80 I am still calculating time/ distance / trajectory. Sharing with others. TRJM
I always look at things graphically first. I saw it as finish the length to 50. Or 2 x 25 = 50 then i squared that to 2500. Then ÷ x 2. 1250. Again for this problem this is the easiest solution , at least in my brain . 25 x 2 = 50 50 x 50 = 2500 2500 ÷ 2 = 1250. Very simple for a square.
Then, I’m also standing at the midpoint of the hypotenuse of two identical, isosceles, right-triangles with a base of 50’ and a height of 25’. Since 1/2xBxH is the area of one then BxH is the area of both = area of square. BxH = 50’x25’ = 1250sq.ft.
You could solve it by multiplying the known length ('25 center to corner) and multiply by itself, (utilizing a perpendicular side) to get the area of half of the square, then multiply by 2.
Rather than having two unknowns, make the problem have one unknown. Draw the square. Draw a line from the top right to the bottom left corner of the square. Draw another line from the top left to the bottom right corner of the square. Call the line from the center of the square to the bottom left corner A. Call the line from the center of the square to the bottom right corner B. And call the bottom line of the square C. Now A squared is 25 squared, or 625 and B squared is 25 squared, or 625. the answered is then 625 +625 = 1250 which is C squared.
Another way to look at it is to say the sq divided by two lines across the center of the sq to the corners makes a 45 degree angle at the corners leaving a 90 degree angle in the center so 50 divided by 2 = 25 and 25 squared = 625 which is half of the area times 2 = 1250 sq ft.(Easier). I see others have come up with the same solution, I couldn't do it in my head though.
Simplest solution is that in the CENTER of the room AND 35 feet from a corner you at the tip of four separate and equal triangles with two sides of 25’. Using A^2 + B^2 = C^2 we know that each side of the square room is 35.355 feet long and so the area of the room is approx = to 1249.98 or 1250 SqFt
Basically boils down to the same idea behind the question to double the size of a square by cutting it into 4 triangles and mirroring all triangles on the squares' edges.
Let's solve in general A square has equal sides Let's say the side is "b" This AREA = b x b or AREA = b^2 Remember that What we do know is that the distance from corner to diametrically opposite corner is x This X^2 = b^2 + b^2 or x^2 = 2b^2 or b^2 = (x^2)/2 But we already know: b^2 = AREA Thus: AREA = (x^2)/2 As we are 25' from each corner x= 50 Thus: AREA = (50^2)/2 or AREA = 2500/2 or AREA = 1250
Divided the hypotenuse in half giving you a right angle triangle which you accurately calculate the sides of the square hence the area 2x the sides equaled 1250
So many ways to solve this. I used cosine 45 = 0.707 X 25 which is 17.675 for half of one side. Double it to give 35.35 for one side. Square that to give area of 1250ft^2
25 x 25 x 2 or 625 x 2 = 1250. >>> draw the diagonals in the square = 4 triangles. Half diagonal (25) x adjacent half diagonal (25) / 2 is the area of 1 triangle so 25 x 25 is the area of two triangles but you have four or twice as many so … 25 x 25 x 2 = 1250
Very easy way to calculate this using a 3,4, 5 ratio triangl the 5 ratio being the hypotenuse . If you are 25 feet away from the corner that means your hypotenuse is 50ft, which means two sides of the room are 30ft by 40ft. So the simple equation is 30ft x 40ft = 1200sqft
Draw a picture and use basic geometry. From your position to a corner, any corner is 25. From one corner to opposite corner is 50. 50 times 50 = 2500 square feet. This is twice the area you need to compute, therefore divide by 2. Answer: 1250 sq ft.
An easier way to do it is to draw two diagonals. each leg will be 25 so a^2+b^2=c^2. 25^2=625 so c^2 = 1250. That is your answer 1250 square feet. You could carry the calculation one more step and get the square root of 1250 for one side of the square but then you would turn around and square one side to find the square feet and it would still come out to 1250.
I had an easier way to do this. If you draw a line from the center to each corner, you get right triangles with lengths of 25'. So ((25' * 25)/2) *4 = ((625)/2)*4 = 321.5 * 4=1250 Or you can simplify the equation by knowing the upper and lower triangles make a square and the left and right triangles also make a square. So (25*25)*2=1250
Very easy.the double of 25ft will be the diagonal of square. So diagonal=50ft Now the formula of diagonal of square is s root 2 where s is the side of square. We alr know diagonal so from this formula we can find side S=50/root 2 we know formula of area of square is s×s so the area is equal to 50/root 2×50/root 2 =2500/2=1250ft. Thanks
I just got the outcome in 3 seconds. It can be solved easier than it is shown. Acutally there are 4 rectangular triangles which legs are equal and have dimension of 25 feet. Then the area of the square is 4*(1/2*a*h) = 4*(1/2*2*2) = 1250.
Diagonal one times diagonal two divided by 2 Or.... 1/2 based times height basis, 50 height is 25. You could just stop right there 'cause you're working with the square
I did it a lot faster. Pythagorean theorem says A² + B² = C². The area is then A×B. Since A=B in a square, it can be simplified as 2A² = 25² 2A² = 2500 A² = 2500÷2 A² = 1250 So the area is 1250.
Hardly anyone realizes it but Pythagoras was one of the Europeans that visited The New World over 2000 years before Columbus. And when he went he took presents for the leaders there. One of the things that most pleased them were hides of animals they could never have imagined. The Chieftains would place these gifts on the ground for their wives to sit on at the ceremonial dinners. One Chieftain was Pythagoras’ favorite and he had given him skins of a Giraffe, a Rhinoceros, and a Hippopotamus. At the ceremonial dinner Pythagoras looked at the Chieftain and his family and a moment of enlightenment washed over him as he saw the wives. Two wives sat on the Giraffe skin, three sat on the Rhinoceros skin and the other five on the Hippo’s skin and he exclaimed, “Eureka! I see it! I understand now! The Squaws of the Hippopotamus is equal to the Sum of the Squaws of the other two Hides!”
@@tedmoss That was a different Chieftain. He was the one that Pythagoras saw and confirmed his new theorem. That chieftain's wives sat 2 on buffalo, 4 on lion, and six on the hippopotamus hide.
1250 square whatevers. Full diagonal is 50 feet. Square it, 2500. The sum of the squares of the two sides is 2500. Thus one of them is 1250. Square root of that is 35.3... which is length of one side but it doesn't really matter since you are going to square it anyway to get area.
Area is made up of 4 right angled triangles 2 sides of which are 25 feet so area of the square is 4 times half base (12.5) times hight of triangle (25) = 1250.
Alternatively you could solve for area of triangle = half base x height. So with 50 as base height is 25. That's 25 x 25 = 625. That's only half the square so double it for the whole square 1250
First read .. ok 25 feet from a corner in a square room = 25 feet from EVERY corner .. so the square room is made up of 2 right hand triangles with an area of each 50 * 25 / 2 .. times 2 triangle = 50*25 ..which is 1250 total area - no calculator needed except a little understanding of how squares and their diagonales relate to each other
that was easy 50 * 50 / 2 Edit: Many moons ago I was exposed to a riddle which I only remember fragmented. A swimming pool should be doubled in size but four trees, one on each corner of the pool must not be moved. The solution was to have the trees in the middel of each side of the new pool. So if the diagonal of a square becomes its side the area doubles. That I deducted from that riddle some 55 years ago.
It’s a 1-1-1.414 triangle times two. Divide 50 by 1.414 to get one side. That’s 35.36. Square that to get the area of the square. That’s 1,250.33 square feet.
My solution doesn't really require more than basic maths, divide square into 4 triangles visually combine opposite triangles to get two squares 25 feet by 25 feet=( 25x25) + (25x25) =1250 there is no need to calculate how long the rooms walls are to answer the question, This took much longer to write than to solve These sorts of questions would would always get a tick and the teachers note "show your working" (
Try X=35.35534 on an 8-digit calculator. It will return exactly 1,250. If you try it on a bigger calculator, you will be over on the 9th digit. (1,250.00006651 on a 12-digit calculator)
25ft at the centre of a square so you have the base and height of 4 triangles that make the total area. 25*25/2 then multiply by 4 triangles. Simplifies to 25*25*2. You actually get the correct answer here rather than having to round.
Did this in my head in about one second. If it's 25 feet to a corner, then a diagonal is 50 feet. So the room can be looked at as a diamond inside a 50 foot square. A diamond has half the area of a circumscribed square. So half of 50 squared is 1250.
I just subscribed to your channel even though I know nothing about math (really) just because I find it entertaining. I did solve the problem (well, not exactly!) though because I have a background in softball. I knew the distance from home to 2nd (84’10” ) and home to 1st (60’) on a softball diamond and converted them to inches. I then (using a calculator) got the ratio figure of .70727 Since your room was 50’ from corner to corner I got 35.3635’ Multiplied that by its self to get 1250.577 I’m sorry to butcher your problem but I really enjoyed it. Thanks.
Solution without Pythagoras' theorem: Cut the square at the two diagonals into four small triangles. Those triangles are kongruent and right-angled. Combining each two of the triangles at their long sides gives you two smaller squares with side length 25 ft. So the area of the large square was 2 * 25 * 25 sqft
i find an easier way to solve this with a different right triangle. if we just draw 25 to each corner the edge becomes the long side (hypotenuse). then 25 squared plus 25 squared is c squared so the edge is c squared at 1250. so the squareroot of 1250 is the edge. so then square that to get the area. (this was easier to do visually without paper but i admit i still used a calculator to get 25 squared times 2)
a square has 90° corners a square has a corner to center line with 45° angle at corner therefore, the triangle from corner to center to mid-side back to starting corner is an isosolese having angles 45, 45, 90. The sides are of proportion: 1:1:sqrt(2). In this problem, 25ft relates to sqrt(2). The other two sides are, therefore: 25/sqrt(2) = 25(sqrt(2))/2 =12.5sqrt(2) This length represents half of the squares side, so that the side's length = 25(sqrt(2)) Area = side×side = (25(sqrt(2)))^2 = 625×2 = 1250ft^2 verify❌️ corner to center =sqrt((25/2)^2+(25/2)^2)❌️ =sqrt(156.25+156.25)❌️ =sqrt(312.5)❌️ =17.68 ❌️=25 Verify pt2: 25sqrt(2)/2)×(sqrt(2) =❤25✔️ review: for an isosolese triangle a^2+a^2=h^2 here h = 25 a^2+a^2=25^2 2a^2=625 a^2= 312.5 a =17.68 for bigger square s=2a =2(17.68) =35.36 area = 35.36^2 = 1250ft^2
You did some extra work, though. Since you knew about √2, you could just start by knowing that the diagonal (50) is √2 times the length of a side of the square. Side of square is 50/√2. Square that for 2500/2 = 1250.
this is easier . 2 diagonals makes 4 r/a isosceles triangles each side 25 ft.. take 2 opposite triangles and put them hypotenuse to hypotenuse and you have a square of 25 ft. side / same with other 2 triangles. 25 squared equals 625 times 2 equals 1250sq.
I've always liked Trig and I went the Trig route:
Sin (45) = .7071
Sin x = opp/hyp
.7071 = opp/25, opp=25(.7071)
opp = 17.67*2 (for the full side of the square)
35.35 * 35.35 (area of a square)
= 1249.811
a little more precise than just 1250, which if you're cutting wood, the .19 is about the width of a saw blade(thin kerf) and can make a difference
Which is why you should do it that way. Less precise because you didn’t use enough decimal places. If you use more decimal places the value will be 1250 exactly.
Do not complain if you don't like this teacher. He is helping people that are learning math, not people that already can math.
Keep on teach, teacher 👍
@borjewahlen6917 I totally agree with you and I will add that he is an excellent teacher.
His solution is really doing it the hard way. A much simpler solution is available.
My issues with this teacher are a) the length of time he takes to explain things, and b) that he does not always demonstrate the easiest solution.
If someone is learning the basics of mathematics, the simpler, or more interesting, you can make it seem, the more likely they are to want to learn more.
And the benchmark of a good teacher is having a student WANT to learn more, not leaving them feeling they need more help, or that it is too complicated to bother with.
@@davek6415, first, a student who is not already familiar with this process or why it works may not absorb it as fast as someone who already understands similar math. He isn't trying to teach those who already grasp it. Second, perhaps he feels that his method is the one that seems to make the most sense, is the most intuitive to a student new to such problems. That is not always the same thing as the easiest process, the fewest steps to obtain the correct answer. A good teacher is someone who understands the psychological processes of their students.
@@johnwest7993 Well, my son, a math teacher agrees with me that the easiest solution is the best solution. Why take the students around the world to solve this ? Imagine taking such a long method on a timed test. You could already be on the fourth problem by the time you work this one out using his method.
You are at the centre of a square, 25' from a corner, so, 25' from each corners. Take any two adjacent corners, and you have a right triangle, with one face of the square as it hypotenuse. So, take 25^2 * 2 to get the square of length of the hypotenuse. So, 25^2 * 2 = 625 * 2 = 1250. You don't need to find the square root of 1250 just to square it. Meaning that you already have the area of the square room : 1250 square feet.
Cool approach. Thanks.
I did it in my head by just knowing that the diagonal of a square is √2 times the length of a side. So 50/√2 is the length of a side. Square that: (50/√2)² = 2500/2 = 1250.
That's exactly how I did it. Seems easier than the method shown in the video.
I did it this way in my head before watching the video - but I build stuff so maybe that helped.
I did the same thing in my head before watching the video. Just visualize it.
Right. I went the same path. It's incredible that they made an 18 minutes video for a question that can be solved in 30 seconds...
Triangle, area of = base times height divided by 2.
You are 25 feet from the 3 corners of a right triangle therefore the base is 50 feet and the height is 25. You have 2 of those triangles therefore the 2’s cancel, so 25 times 50 is 1250.
A second way is to recognize that a square is comprised of four right triangles that form two smaller squares that have a side length of the distance from the center of the larger square to one of its corners.
25*25= 625 is the area of each small square times 2 = 1250 sq ft.
A) youngster, I enlisted in 1971 and retired in 1995 from the Navy.
B) I fold the flag when we bury one of Uncle Sam’s Misguided Children
(For those of you that are confused, there is branch rivalry, however we band together when an outsider gets involved.)
The diagonal of the square room is 50ft corner to corner. Therefore 2 x side^2 = 50^2. side^2 = area = 50^2/2 = 1250 sqft
That's what I did in my head. 🤷♂️
@@Gideon_Judges6 Yup. Not difficult.
It took me less than 20 seconds to read your explanation. I should have gone straight to your comment. Sadly, I watched John’s 18 minute lecture first. It was, how shall we say, an ordeal.
That's how I did it.
That's what I did.
Easier mental math - from the center to the corner is 25 and also to all corners. So, From the center to two corners forms an isosceles right triangle which is 1/4 of the square. Base times height is 25*25 or 625. The square is twice that value (or (625/2) * 4), or 1250.
@@walkfaster Ok, here's another way. Slide rule enthusiasts should not be left out of the fun.
If you're 25 feet from the corner, the area of the square is 10^(log(25)*2+log(2)) = 1250.
log(25)*2 is the log of 25^2. log(25^2)+log(2) is the log of 25^2 times two.
You raise ten to that power because the base ten log of a number is the exponent you need to use against a base of 10 to get the number.
Note that any log base will do, so e^(ln(25)*2+ln(2)) also equals 1250.
However, log of 25 is hard and any log will do, right?
How about 25^(2+log(2)/log(25))? That also equals 1250, but is trickier.
2 is log base 25 of twenty five times 2, because log base 25 of 25 is 1. 1*2, Terrance Howard notwithstanding, is 2.
log(2)/log(25) is the base 25 log of 2. That converts the base ten log of two to its base 25 counterpart. Addling logs is like multiplying the bases.
Adding it to the log_25(25)*2 is, again, multiplying by 2. You get the base 25 log of 25^2 * 2.
Take 25 to that power and you get 1250.
Now, to my calculator to see if that worked. I'm pretty sure it did.
@@walkfaster You have impeccable taste!
@@johnnyragadoo2414 I had a Pickett, back in the 1960s.
@@darkdelta A EE professor I had was catching grief from a computer major who thought computers would soon obsolete engineering. “Young lady,” he said, “we put men on the moon and returned them safely with slide rules and three digits of precision.” I will never forget that comeback. That was Professor Dougal at UT Austin. An amazing guy.
@@johnnyragadoo2414 Shut herdown pretty ole quick.
I can remember talking with some colleagues, again in the 1960s, how one day we'll get electronic adding machines. We never considered multiplication and division. Then Burroughs came out with a desktop calculator, it sported a Nixie tube display, among other features, Now my phone has more computing power than some mainframes back then,
At age 76, I worked this out in my head. The 4 isosceles triangles form two squares with side 25. So 2 x 25 x 25 = 50 x 25 = 100 x (25/2) = 100 x 12.5 = 1250
Construct your mental box at 45 degrees to your illustration In short 50X50 simple 2500 square feet, half of that is 1250. Yes I did Pythagorean Theorem as well to confirm,, and I did the hypotenuse of a 45 degree triangle is A X 1.414. 50' divided by 1.414 is about 35 (plus a little) and 35X35 (plus a little) is about 1250 Three methods. The 50X50 divided by 2 is by far the fastest easiest.
Length of a diagonal of a square= a✓2 (where a is the side of the square)
As such , a✓2 = 50
a = 50/✓2 ft
Area of the square= a^2
= (50 ft/✓2)^2
1250^ft^2/2= 1250 ft^2
My way also.
best reasonning answer.
OMG! Between the poster's long-winded, repetitious verbalizing and the commenters, I have to say that I now understand why America is 37th in the world in education. I was a not a good math student in school (now 75!) but I had this figured out almost in my head in a few minutes. Too much talk and too much complicated explanations that didn't explain the basic "need to know", i.e. a sq + b sq=c sq, and c=25, sq = 625. a and b are equal, since the room is square, and a sq is half, or 312.5, and b sq is half, the sq root of which is 17.6776, so the area of one of 8 triangles (base times height) is 17.6776 squared=312.5/2 (or a + b)=156.250 x 8 (triangles) =1,250. Some of the explanations in the comments make no sense, and by the time the poster was done yapping about incidentals, I had already stopped listening. That is the problem in school! It was the problem 60 years ago when I was in jr. high. God help our children!
Almost in your head....means you used caculator. I would have needed a calculator too. Some of these commentors are suggesting different ways to find the solution so they are adding value. Had I not skipped to comments I would have used method described on video.
It's a 45/90 triangle, so one leg is hypotenuse/sqrt(2) = 50/sqrt(2). Don't have to calculate what that is. Area is side^2, so 2500/(sqrt(2))^2 or 2500/2=1250. No calculator.
@@PaulTaylor-Young To be fair to the guy who posted the video, in a math test they probably want to know that you know all of it. In the real world there are definitely easier and quicker ways to do it, but all rely on you knowing the basics and being able to visualise it.
Draw the diagonal of the large square. The diagonal will be twice the distance from you to the corner, so 50 feet. This divides the square into 2 triangles. Choose either triangle and construct a square on each side of it. The square on the hypotenuse is 50 x 50 = 2500 square feet. By the Pythagorean theorem we know that the sum of the other two squares is also 2500. Further, the sides of the smaller squares will be equal, so their areas are also equal, and will be 2500 / 2 = 1250 square feet. Now notice that the original square had side length exactly the same as the sides of the two smaller squares. Therefore the area of the original square is exactly the same at 1250 square feet.
1250.32
@@sutters7251 What?
Trick question. 25 metres from "a" corner, not all corners. Can't be answered.
@@raamannair8072 Um, just before that it says you are in the *center* of a *square* room. Since this is a geometry problem, it is reasonable to interpret “square” as geometrically square, and “center” as the literal geometric center, so the corners are equidistant from you. If this were a geometry test you’d get zero marks for not demonstrating that knowledge. If I were marking it you might get -1 for being pedantic and actively denying this geometric truth.
Without watching the video the easiest way I found was to create a triangle using two sides going to corners with a side of the square being the hypotenuse. So the length of the side of the square is x^2 = 25^2 + 25^2. This becomes x2 = 1250. You can really stop there are the area of a square id L x B. And, in this case X^2 = L x B = 1250.
Like a lot of people out there I’m old and retired. Been too many years to remember this stuff. I ‘m happy for a chance to study these again. Love your explanations. Hopefully it will help keep my mind sharp
We are so impatient that we tend to forget that we are not the only one in the room. I already know most of what is discussed on this Channel, but I realize that he is catering for the whole class. It's the mark of a good teacher. He makes it easier for the slower one. Let's therefore remember those whose struggle with these concepts.
As a teacher myself, I have learned that other people (students) explaining their methodology can ‘click’ for some students who don’t get the teacher’s explanation. ** Even if they sometimes use the exact same words. 🤪
d^2/2= A. D= diagonal. A= area. The diagonal is twice as long as 25 feet. That is 50 feet. 50^2= 2,500 feet. 2,500 ÷ 2= 1,250.
It's also 2 right triangles with base of 50ft and height of 25 ft.
2 • 1/2 • b • h (but 2•1/2 = 1)
b • h = 50 × 25 = 1250 ft^2
Only one calculation necessary!
I looked at it like four triangles.
If you put two of the triangles together you have a square with sides 25 x 25.
25 x 25 = 625 Sq ft
625 x 2 = 1250 Sq ft
I solved it the same way. Took me 10 seconds.
If a person doesn’t know the method of figuring out the area of a triangle, there’s perhaps an easier way to solve this: if you start at a corner and walk to the middle of the room, you’ve gone 25 feet; if, instead of continuing on to the _far_ corner, you turn right and go to the _next_ corner, you’ve gone another 25 feet; if *now* you imagine that you’ve just walked down two adjacent sides of an imaginary square that’s half in the room and half outside the room, then that square is 25 by 25, only half of which is in the room; 25 by 25 equals 625; you can divide that in half to get the area of the imaginary square that’s in the room, *but* , since the area of the imaginary square that’s _in_ the room takes up only one quarter of the room, we can imagine a second imaginary square of the same size that takes up another quarter of the room - in other words, don’t divide the 625 in two - - instead, we see that 625 square feet is the area of exactly *_half_* the room; so, simply double 625 to 1,250 and you now have the total area of the room.
This method and explanation started out good, but "imaginary squares" weren't needed. Just calculate the area of that right triangle with base and height 25. Multiply by four. Or multiply by four first so you don't need to calculate 625/2.
The square room can be thought of as 4 right triangles each with a base = height = 25.
Area of each triangle is 1/2 × 25 × 25
Since there are 4 such triangles. Total Area of Room is
4 × 1/2 × 25 × 25 =
2 × 625 = 1250 FT²
So for any Square Room with the distance given from corner to center or vice-versa the area of the square is that distance given ^2 (squared) and then doubled.
So if you are in the center of a square room 10 FT from the corner. The area of the room is 2 × 10^2 = 2 × 100 = 200 FT²
Or, the really easy way to solve it; 2 x 25 x 25 as you're in the centre of a square and 25 from the corner, that means that you can divide the room into 4 diagonally and make 2 squares of side length 25. Thus if each square has a side of 25 then twice the square of 25 is your answer. Which is 1250 ft squared.
For a unit square, the sides are 1 inch, and the diagonal is the square root of 2. All squares will have this same ratio. Therefore, if a square has a diagonal of 50, its sides are 50 divided by the square root of 2. The square of that is the area.
1/2 x base x height = area of a triangle. 1/2 x 50 x 25 = 625. We have two such triangles. 625 + 625 = 1,250 ft^2.
Good idea, but base and height is 50 / sqrt (2)
1250, straight forward application of fundamental procedures.
Form your diagram, Area of a triangle = 1/2 * height (h). Base =50, h = 25, but area of a square = 2* area of triangle. So area = 2. (1/2.50). 25= 1250 sq feet. OR. From your diagram, x= side of square. So, by Pythagorus, x(sq) = 25(sq) + 25(sq). But area of the square = x(sq). So area of square = 2(25(sq)) = 2(625) = 1250 sq feet. I appreciate these 2 solutions are linked as they both give 2(25.25) = 1250 sq feet..
The hypotenuse is 50 feet as you are in the center of a square room, this also means that the length = width (a = b)
a^2 +b^2 = 50^2
a^2 + a^2 = 2500
2a^2 = 2500
2a^2/2 = 2500/2
a^2 = 1250
(sqr rt) a^2 = (sqr rt)1250
a = (sqr rt) 1250
a & b = 35.36 ft (rounded to the nearest 100th)
a*b = 1250 ft^2
Just because it's a square, it doesn't mean you need to figure out square roots. If you are 25 feet away from a corner, and you are in the center, then you can divide the square into two or four triangles. The area of a triangle is 1/2 the base times the height. 25 is the base and the height for a quarter triangle. 25 times 25 is 625. that is two quarter triangles, which is half of the square. 625 plus 625 is 1250. You can also do two triangles, which is easier. 25 + 25 is 50. 50 is the base of a half triangle. 25 times 50 is also 1250. You don't need to halve the answer, because there are two triangles, when you bisect the square from corner to corner.
I just turned the room into two equal triangles. .5 × base × height is all you need for the area of a triangle. So. .5 × 50 ×25 = 625 then double that to 1250 since two triangles make the square.
Needlessly convoluted answer
Area of triangle : (25x25)x0.5
4 triangles = area of square= 625x0.5x4=625x2=1250 QED
Er, if you are in the centre then you are a distance “a” from each side and the length of each side is 2a so the area is 4a^2. Using Pythagoras, applied to the triangle formed from the 25ft diagonal from you to the corner and the distance “a” from you to each side shows 2a^2=25^2=625. So if the area is 4a^2 then the area is 625 x 2 = 1250ft sq.
I did it by finding the area of an interior right triangle with side lengths of 25, then multiplying by 4, because you have 4 of those interior right triangles and the combined area of those 4 interior triangles equals the area of the square room. Thus, the area of one of the triangles = 1/2 * (25*25) = 1/2 * (625) = 312.5, so then 312.5 * 4 = 1,250 sq. ft. for the area of the square room.
What a long winded way of doing it! Just form the right angle in the centre, by going to an adjacent corner, not the opposite one, so you have a triangle one quarter of the square’s area. The area of this is simple: area of a triangle is half base x height, in this case 12.5x25. But as there are 4 such triangles, the area of the square is therefore 12.5x25x4, or 12.5x100, ie 1250. No need for Pythagoras’ theory etc
I used Pythagoras and area of square formulas but made the length of one wall the hypotenuse. Center to each corner is 25. So 25^2+25^2=625+625 so one wall is 1250. Square root if 1250 is 35.355 for one wall. Area = LxW or 35.355x35.355 or 1250.
To the corner there are 25 feet. That's half the diagonal. We can use that diagonal in the formula.
(50ft)^2/2=1250 ft^2
(the whole process was to do 25, *2, ^2, /2, = .)
If we don't remember that formula we can easily get there with Pythagoras:
50^2=side^2 +side^2
2500=2*side^2
1250=side^2
Another option is to think that those 25ft semi-diagonals trace 4 right triangles inside the square.
25ft*25ft/2=312.5ft^2
*4
=1250ft^2
25ft to the centre, so diagonally corner to corner is 50ft. Two sides of the room and that diagonal form a right angle triangle where the other two angles are 45 degrees. The hypotenuse of such a triangle is √2 times the length of each of the other sides (you can check that with Mr Pythagoras) so the side length of the square room is 50/√2.
Therefore the area = (50/√2)² = 2500/2 = 1250 ft²
That's exactly how I did it. This is the second video of John's that I have seen where he did not use the fact that the diagonal of a square is √2 times the length of a side. I think he just wants to show the Pythagorean steps, which is just fine.
Diagonals in a square intersect at 90 dgr. So a side of the square can be the hypotenese of a right-angle triangle with equal cathetuses
Two sides of a triangle, equal to each other, with the hypotenuse being third side. Each side is25. So 25 squared plus 25 squared equals the hypotenuse squared, which is the area of the square.
Pythagoreas. a^2+b^2=c^2. In this case a=b for the sides of the room. The diagonal of the square is c, which is double of 25 = 50. c^2=2a^2 = 2(area) 50*50/2 = area. 2500/2 = 1250. Did it all in my head.
50^2 = 2X^2 (Pythagoras) => 2X^2 = 2500 => X^2 = 1250. The area = X x X = X^2 = 1250 (feet^2, in this case.) No need to solve for X here, given the fact that it is a square room.
@@sinnirr X = sqr. rt. X^2 = sqr. rt. 1250 = 25 x sqr. rt. 2. This is of course the length of each side of the square (of course a positive number).
@@Kleermaker1000 Yeah, I was more interested in the sides, not the area... I meant to delete my dumb answer.
@@sinnirr It wasn't a dumb answer, but now you know the sides of the square. :)
@@Kleermaker1000You actually need to solve for X. Because you're in a square, and your distance is from the corner, not the middle of an edge. Need to multiply your 25' by sin (45) = 41.545
Square that, gets you 1810.078 sf
EDIT - Aw hell. My calculator was on radians for some reason. Changed to degrees and wouldn't you know it, 1,250sf
Yeah. I had to check my math several times because I wasn’t sure it was that simple. But it is.
The answer is firstly, one has to figure the length of one of the sides of the square. One knows that one is in the middle of the square where one is standing 25 units from a corner. Now, this is actually 50 units from corner to corner and that this length is 45 degrees at a corner. So, one uses the formula cos theta = x/h , base/hypoteneuse. So, x as one of the legs of the square is
x = cos(45 degrees) * 50 and this gives you the leg of the square. Area of the square then becomes ((cos45 degrees)*50)^2.
I made a small right triangle in my head that had a hypotenuse of 25 ft. I ended up with sqrt(625/2) ft. for each triangle side. Therefore, multiplying this by 2 to get the length of each side of the big square (same as multiplying everything inside the sqrt by 4), you get sqrt(625 × 4 / 2) or sqrt(625 × 2) or sqrt(1250), and, because the area as this times itself, that just undoes the sqrt and the area is 1250 sq. ft. But, yes, your way was a bit quicker.
A rather easy solution to this problem is if you’re in the center of a room and 25 feet from the corner and the area of a triangle is 1/2 the length times the height then
25×25=625
Now you divide that in half
625÷2 =312.5
And now you have the area of a triangle, that is equivalent to 1/4 the area of the total of the square which means multiply times four and you get the area.
312.5x4=1250
Of course, if you’re thinking ahead of the game then 625 would equal half the area of the room because it’s two of your triangles that would make up the total of the area of four triangles so you could just multiply that by two to get the same answer. Sounds more complicated than it really is. But what I’m basically saying Is calculate the area of a triangle where a side of the square is the hypotenuse of a right triangle where the two sides are 25 feet which is equivalent to 1/4 of the total area of the square.
Except stop after 25x25=625 because you have the area of the triangle that is one half of the square. (Corner 1 to corner 2 to corner 3.) Just multiply that by 2.
@@thomasharding1838 yeah I said that which makes me think you didn’t read everything I wrote.
It is a right, equilateral triangle and an hypotenuse of 50. A 1, 1 similar triangle has a hypotenuse of square root(2), or remember this relationship from prior interactions with this triangle. If we chose a similar triangle with a hypotenuse of 50, the sides are the 50 divided by the square root of 2. The square’s area is one side squared or 50 squared divided by the square root of 2 squared or 2500/2. So area is 1250 square units.
If you're in the middle of the room then the hypotenuse is the length of 25'' therefore A squared + B squared = C squared or 625'. A = B so A squared X 2 = 625'. The room is approximately 17' 8" X 17' 8" (17.6776695). 17.8 X 2 = 619.52
@16:40 Keep in mind that X is only approximately equal to 35.35 ft but it is exactly equal to the square root of one half the square of the hypotenuse. The area of a SQUARE is equal to one half of the square of the hypotenuse and will equal the square of one side.
Square of the hypotenuse divided by 2. The lesson to take away is that you don't always need to know the side length to work out an exact area of a square.
The room can be divided into two triangles with a base of 50' and a height of 25'.
1/2 bh*2=bh=50'*25'=1250'.
The diagonal is 50 but we want a side length to find the square's area, so draw another square around this one (in your head of course) turned 45 degrees so that the corners of our square are in the middle of the sides of the new one. We have the side length for the new square as a given (near enough) 50'. So the area of that square is 2500 sq ft, and it is twice the size of our original square so our square is 1250 sq ft.
Takes much longer to write than it does to work out.
Easy mental arithmetic.
Area of one triangle here with 50' hypoyhenus and the height from the hypothenus to the corner, is 25. Area of the triangle is then 625, and the area of the full square is 1250.
Alternately: sin (a)= opp/hyp thus: hyp * sin (a)= opp which leads to 50(sin 45°) = 35.35 ^2 = 1250sq ft where a +45°
1250 ft^2
Two triangles with base = 50 & ht = 25
A = 1/2 • b • h • 2 = b • h
A = 50 • 25 = 1250 sq ft
A simpler way. Knowing that from center to two adjacent corners of a square forms a right triangle with each line equidistant and has the side S of the square as the hypotenuse. Let's call this line C which is given as 25ft. So C^2 + C^2 = S^2. 25^2 + 25^2 = S^2 = 625 + 625 = 1250sqft. Since the area of the room is S^2 we already solved our problem.
I knew that hypotenuse of 45 degree triangle was 1.41412 so I divided it into 50 giving 35.357. Then multiplying the side squared = 1250.141. Area. If it had been a 60/30 triangle or rectangle with relationship of hypotenuse 1.732 into what ever number was the center of the rectangle. Your way was my first as the numbers were arrived without calculating the length of sides. However in any test I ever took 2 to 3 components were derived from initial answer to first part, resulting in being critical to calculate the correct answer for part one of the details reflecting each unknown measurement of the object. In this case, what if the square became a cube and volume requested. Excellent find, I will pass this on! At 80 I am still calculating time/ distance / trajectory. Sharing with others. TRJM
I always look at things graphically first. I saw it as finish the length to 50. Or 2 x 25 = 50 then i squared that to 2500. Then ÷ x 2. 1250. Again for this problem this is the easiest solution , at least in my brain . 25 x 2 = 50 50 x 50 = 2500 2500 ÷ 2 = 1250. Very simple for a square.
I used the side of the square as the hypotenuse and the two sides are 25 feet. Giving me 2 (25E2) which is 1250 SQFT.
Then, I’m also standing at the midpoint of the hypotenuse of two identical, isosceles, right-triangles with a base of 50’ and a height of 25’. Since 1/2xBxH is the area of one then BxH is the area of both = area of square.
BxH = 50’x25’ = 1250sq.ft.
You could solve it by multiplying the known length ('25 center to corner) and multiply by itself, (utilizing a perpendicular side) to get the area of half of the square, then multiply by 2.
Rather than having two unknowns, make the problem have one unknown.
Draw the square. Draw a line from the top right to the bottom left corner of the square. Draw another line from the top left to the bottom right corner of the square. Call the line from the center of the square to the bottom left corner A. Call the line from the center of the square to the bottom right corner B. And call the bottom line of the square C. Now A squared is 25 squared, or 625 and B squared is 25 squared, or 625. the answered is then 625 +625 = 1250 which is C squared.
Did this one in my head...
Another way to look at it is to say the sq divided by two lines across the center of the sq to the corners makes a 45 degree angle at the corners leaving a 90 degree angle in the center so 50 divided by 2 = 25 and 25 squared = 625 which is half of the area times 2 = 1250 sq ft.(Easier). I see others have come up with the same solution, I couldn't do it in my head though.
Side calculated as: (2*25) / √2
50/1.414= 35.36
Area of a square is S*S
35.36*35.36=1,250.33
Simplest solution is that in the CENTER of the room AND 35 feet from a corner you at the tip of four separate and equal triangles with two sides of 25’.
Using A^2 + B^2 = C^2 we know that each side of the square room is 35.355 feet long and so the area of the room is approx = to 1249.98 or 1250 SqFt
Basically boils down to the same idea behind the question to double the size of a square by cutting it into 4 triangles and mirroring all triangles on the squares' edges.
Let's solve in general
A square has equal sides
Let's say the side is "b"
This AREA = b x b
or
AREA = b^2
Remember that
What we do know is that the distance from corner to diametrically opposite corner is x
This
X^2 = b^2 + b^2
or
x^2 = 2b^2
or
b^2 = (x^2)/2
But we already know:
b^2 = AREA
Thus:
AREA = (x^2)/2
As we are 25' from each corner x= 50
Thus:
AREA = (50^2)/2
or
AREA = 2500/2
or
AREA = 1250
Did this by thinking of 2 identical triangles splitting the square. Area of triangle = 1/2 b X h. b = 50, h = 25 so 25 X 25 X (2) = 1250
Divided the hypotenuse in half giving you a right angle triangle which you accurately calculate the sides of the square hence the area 2x the sides equaled 1250
I think of it this way. Four right angle triangles 25x25 is the same as two full squares 25x25. So 625+625 is 1250.
So many ways to solve this. I used cosine 45 = 0.707 X 25 which is 17.675 for half of one side. Double it to give 35.35 for one side. Square that to give area of 1250ft^2
25 x 25 x 2 or 625 x 2 = 1250. >>> draw the diagonals in the square = 4 triangles. Half diagonal (25) x adjacent half diagonal (25) / 2 is the area of 1 triangle so 25 x 25 is the area of two triangles but you have four or twice as many so … 25 x 25 x 2 = 1250
Very easy way to calculate this using a 3,4, 5 ratio triangl the 5 ratio being the hypotenuse .
If you are 25 feet away from the corner that means your hypotenuse is 50ft, which means two sides of the room are 30ft by 40ft.
So the simple equation is 30ft x 40ft = 1200sqft
Divide the square into two triangles, as you did, then it’s just base x height to get the area of the square. 50 x 25 = 1250
Draw a picture and use basic geometry. From your position to a corner, any corner is 25. From one corner to opposite corner is 50. 50 times 50 = 2500 square feet. This is twice the area you need to compute, therefore divide by 2. Answer: 1250 sq ft.
An easier way to do it is to draw two diagonals. each leg will be 25 so a^2+b^2=c^2. 25^2=625 so c^2 = 1250. That is your answer 1250 square feet. You could carry the calculation one more step and get the square root of 1250 for one side of the square but then you would turn around and square one side to find the square feet and it would still come out to 1250.
I had an easier way to do this.
If you draw a line from the center to each corner, you get right triangles with lengths of 25'.
So ((25' * 25)/2) *4 = ((625)/2)*4 = 321.5 * 4=1250
Or you can simplify the equation by knowing the upper and lower triangles make a square and the left and right triangles also make a square. So (25*25)*2=1250
Very easy.the double of 25ft will be the diagonal of square. So diagonal=50ft
Now the formula of diagonal of square is s root 2 where s is the side of square. We alr know diagonal so from this formula we can find side
S=50/root 2 we know formula of area of square is s×s so the area is equal to 50/root 2×50/root 2 =2500/2=1250ft. Thanks
I just got the outcome in 3 seconds. It can be solved easier than it is shown. Acutally there are 4 rectangular triangles which legs are equal and have dimension of 25 feet. Then the area of the square is 4*(1/2*a*h) = 4*(1/2*2*2) = 1250.
Triangle area = ½bh, square = 2 triangles. 2•½•b•h = bh = 50•25 = 10•5•25= 10•125=1250. No calculator required.
Diagonal one times diagonal two divided by 2
Or.... 1/2 based times height basis, 50 height is 25.
You could just stop right there 'cause you're working with the square
I did it a lot faster. Pythagorean theorem says A² + B² = C².
The area is then A×B. Since A=B in a square, it can be simplified as 2A² = 25²
2A² = 2500
A² = 2500÷2
A² = 1250 So the area is 1250.
Hardly anyone realizes it but Pythagoras was one of the Europeans that visited The New World over 2000 years before Columbus. And when he went he took presents for the leaders there. One of the things that most pleased them were hides of animals they could never have imagined. The Chieftains would place these gifts on the ground for their wives to sit on at the ceremonial dinners. One Chieftain was Pythagoras’ favorite and he had given him skins of a Giraffe, a Rhinoceros, and a Hippopotamus. At the ceremonial dinner Pythagoras looked at the Chieftain and his family and a moment of enlightenment washed over him as he saw the wives. Two wives sat on the Giraffe skin, three sat on the Rhinoceros skin and the other five on the Hippo’s skin and he exclaimed, “Eureka! I see it! I understand now! The Squaws of the Hippopotamus is equal to the Sum of the Squaws of the other two Hides!”
Unlikely, since the chief had 12 wives.
@@tedmoss That was a different Chieftain. He was the one that Pythagoras saw and confirmed his new theorem. That chieftain's wives sat 2 on buffalo, 4 on lion, and six on the hippopotamus hide.
(50/√2)^2. For any square, side * √2 = diagonal.
45 degree Triangle relationship is 1 ,1 square root of 2 Take 25 divide by 1.414 then multiply times 2
1250 square whatevers. Full diagonal is 50 feet. Square it, 2500. The sum of the squares of the two sides is 2500. Thus one of them is 1250. Square root of that is 35.3... which is length of one side but it doesn't really matter since you are going to square it anyway to get area.
116.1288 Square Metres.
Area is made up of 4 right angled triangles 2 sides of which are 25 feet so area of the square is 4 times half base (12.5) times hight of triangle (25) = 1250.
Alternatively you could solve for area of triangle = half base x height. So with 50 as base height is 25. That's 25 x 25 = 625. That's only half the square so double it for the whole square 1250
First read .. ok 25 feet from a corner in a square room = 25 feet from EVERY corner .. so the square room is made up of 2 right hand triangles with an area of each 50 * 25 / 2 .. times 2 triangle = 50*25 ..which is 1250 total area - no calculator needed except a little understanding of how squares and their diagonales relate to each other
that was easy 50 * 50 / 2
Edit: Many moons ago I was exposed to a riddle which I only remember fragmented. A swimming pool should be doubled in size but four trees, one on each corner of the pool must not be moved. The solution was to have the trees in the middel of each side of the new pool. So if the diagonal of a square becomes its side the area doubles. That I deducted from that riddle some 55 years ago.
It’s a 1-1-1.414 triangle times two. Divide 50 by 1.414 to get one side. That’s 35.36. Square that to get the area of the square. That’s 1,250.33 square feet.
1200sqft. 30’ wide x 40’ long
3,4,5 right triangle
My solution doesn't really require more than basic maths, divide square into 4 triangles visually combine opposite triangles to get two squares 25 feet by 25 feet=( 25x25) + (25x25) =1250 there is no need to calculate how long the rooms walls are to answer the question,
This took much longer to write than to solve
These sorts of questions would would always get a tick and the teachers note "show your working" (
Try X=35.35534 on an 8-digit calculator. It will return exactly 1,250. If you try it on a bigger calculator, you will be over on the 9th digit. (1,250.00006651 on a 12-digit calculator)
You can also sine and cos 45 and hypotenuse as 25 ft
I mean 50
You can also use pythagorean theorem and a=b and c =50 to get one side
Exactly how I did it in under a minute. Thanks for the challenge.
d/2 = 25 ft, area = d^2/2 square ft
= 625 /2 square feet
25ft at the centre of a square so you have the base and height of 4 triangles that make the total area. 25*25/2 then multiply by 4 triangles. Simplifies to 25*25*2. You actually get the correct answer here rather than having to round.
Did this in my head in about one second. If it's 25 feet to a corner, then a diagonal is 50 feet. So the room can be looked at as a diamond inside a 50 foot square. A diamond has half the area of a circumscribed square. So half of 50 squared is 1250.
Each side S of the room is clearly ( [Root-2] / 2) [ 50 ] ft, so that Area = S^2 = 1250 ft^2
I just subscribed to your channel even though I know nothing about math (really) just because I find it entertaining. I did solve the problem (well, not exactly!) though because I have a background in softball. I knew the distance from home to 2nd (84’10” ) and home to 1st (60’) on a softball diamond and converted them to inches. I then (using a calculator) got the ratio figure of .70727
Since your room was 50’ from corner to corner I got 35.3635’
Multiplied that by its self to get 1250.577
I’m sorry to butcher your problem but I really enjoyed it. Thanks.
So many ways to solve but depending on the setting e.g in an examination setting speed should be considered!
Solution without Pythagoras' theorem: Cut the square at the two diagonals into four small triangles. Those triangles are kongruent and right-angled. Combining each two of the triangles at their long sides gives you two smaller squares with side length 25 ft. So the area of the large square was 2 * 25 * 25 sqft
i find an easier way to solve this with a different right triangle. if we just draw 25 to each corner the edge becomes the long side (hypotenuse). then 25 squared plus 25 squared is c squared
so the edge is c squared at 1250. so the squareroot of 1250 is the edge. so then square that to get the area.
(this was easier to do visually without paper but i admit i still used a calculator to get 25 squared times 2)
a square has 90° corners
a square has a corner to center line with 45° angle at corner
therefore, the triangle from corner to center to mid-side back to starting corner is an isosolese having angles 45, 45, 90. The sides are of proportion:
1:1:sqrt(2).
In this problem, 25ft relates to sqrt(2). The other two sides are, therefore:
25/sqrt(2) = 25(sqrt(2))/2
=12.5sqrt(2)
This length represents half of the squares side, so that the side's length = 25(sqrt(2))
Area = side×side
= (25(sqrt(2)))^2
= 625×2
= 1250ft^2
verify❌️
corner to center
=sqrt((25/2)^2+(25/2)^2)❌️
=sqrt(156.25+156.25)❌️
=sqrt(312.5)❌️
=17.68 ❌️=25
Verify pt2:
25sqrt(2)/2)×(sqrt(2)
=❤25✔️
review:
for an isosolese triangle
a^2+a^2=h^2
here h = 25
a^2+a^2=25^2
2a^2=625
a^2= 312.5
a =17.68
for bigger square
s=2a
=2(17.68)
=35.36
area = 35.36^2
= 1250ft^2
You did some extra work, though. Since you knew about √2, you could just start by knowing that the diagonal (50) is √2 times the length of a side of the square.
Side of square is 50/√2. Square that for 2500/2 = 1250.
this is easier . 2 diagonals makes 4 r/a isosceles triangles each side 25 ft.. take 2 opposite triangles and put them hypotenuse to hypotenuse and you have a square of 25 ft. side / same with other 2 triangles. 25 squared equals 625 times 2 equals 1250sq.