Elegant way to find the Perimeter of a right triangle | (step-by-step explanation) |
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- Опубліковано 27 жов 2024
- Learn how to find the Perimeter of a right triangle when two sides are unknown. One side of the triangle is 89. Important Geometry and Algebra skills are also explained: Pythagorean theorem; algebraic skills. Step-by-step tutorial by PreMath.com
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• Elegant way to find th...
Elegant way to find the Perimeter of a right triangle | (step-by-step explanation) | #math #maths
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❤❤❤ thanks 💯🙏 keep going my dear teacher ❤️
Thank you, I will ❤️
You are awesome. Keep it up 👍
The important extra information which is not emphasised is the requirement that sides must be positive integers. If sides can be any positive real number, there are an infinity of answers.
You are correct. If the sides can be any positive real number, there are infinite answers
So the answer provided is not actually the answer to the question as actually posed. An answer but not the answer
I was thinking the same thing!
yes now I got it - as was concluding that there are infinite number of solution as it depends on angle c which can be any between >o
The confusion would have been avoided if it was stated from the start that all sides are positive integers.
We need not find the values of a and c seperately, as the question is 'What is the perimeter? ' Perimeter is a + b + c we have got the value of a + c = 7921, just add a (89) to this to get the perimeter. ( a + c ) + b = a + b + c = 7921 + 89 = 8010, which is the answer you got by finding the values of a and c.
Sorry but without any 2nd side or an angle , there are an infinite number of triangles.
With the given information there are endless solutions. When a nears 0 , c nears 89+ . When a nears endles, c nears endles
Not really - sides have to be positive integers and there is only one solution.
It's not,
Since it's already stated that the side lengths must be positive integers.
Find out 89^2=7921, decide the no into 2 consecutive nos
89^2=3960+3961, as per vedics,89^2= 3960^2+3961^2 implies all the 3 are sides, area is dead easy
Mukund
This problem is incorrectly posed. If you move the point C either left or right the sides 'a' and 'c' will change and with them the perimeter. The problem is still solvable by making an additional assumption, which you actually do when you assign the values.
0:56 “Keep in your mind that the side lengths must be a positive integer”, See diagram: Sides ∈ Z+
It is arbitrary to say that, if xy = zt, then x=z andy=t.
As a matter of fact, there are infinite triangles having a side = 89
It is easy enough to prove your statement - just give us as least one more solution.
yes there are, but the sides must be integer numbers, and the only solution to that is the one that is shown on the vid.
0:56 “Keep in your mind that the side lengths must be a positive integer”, See diagram: Sides ∈ Z+
This solution only works if you assume all values are integers, which was not given as a condition.
Introduce fractions, and there are an infinite number of possible solutions.
Z^+ was given.
For any odd number n greater than 1, there is a Pythagorean triple (n, m, m + 1) where m = ½ (n² − 1).
When n is a prime number, there is no other Pythagorean triple than this one and the perimeter is n² + n.
@@pluisjenijn to be exact, the funny property is n² + m² = (m + 1)²
like (21, 220, 221) (201, 20200, 20201) (2001, 2002000, 2002001)
This is very good to know. For our PreMath problem above, are we just limited to Pythagorean triples? Or could PreMath's solution apply to all right triangles if missing two side lengths? Thank you!
I arrived at the same result because for any prime number b, the second scenario always leads to a=0.
Only one solution is therefore possible for the perimeter p with c=(b²+1)/2 and a=(b²-1)/2
p = a+b+c = (b²-1)/2+b+(b²+1)/2 = (b²-1+2b+b²+1)/2 = (2b²+2b)/2 = b²+b
We can deduce that for each prime number b, there exists a Pythagorean triplet (a, b, c) of non-zero natural integers verifying the Pythagorean relation a²+b²=c² with c=(b²+1)/2 and a=(b²-1)/2!
@@sail2byzantium since the _sides_ ∈ ℤ⁺ (as shown in the upper right corner of the video) we are limited to Pythagorean triples.
But there could be multiple solutions: when b = 33 the solutions are (33, 44, 55), (33, 56, 65),, (33, 180, 183) and (33, 544, 545).
@@sail2byzantium Hello, when PreMath states the solutions are limited to those triangles with sides that are integers he is indeed limiting the answers to Pythagorean triples. And as @ybodoN alertly points out, if the given side is an odd prime number greater than 1 there will be one and only one Pythagorean triple solution.
3-4-5, 5-12-13 and 7-24-25 are the three smallest Pythagorean triples where the the smallest side is listed first. There appears to be a pattern. That is c = b+1. The hypotenuse is one larger than the longer leg. Using a = 89, b, c = b+1, the Pythagorean Theorem and some algebra, you get b = 3960 and c = 3961. P = sum of three sides = 8010.
6.8.10 not like that
b=89 is a prime number
In fact for any prime number b, the second scenario always leads to a=0.
Also there is only one possible solution: c=(b²+1)/2 and a=(b²-1)/2
And a perimeter p = a+b+c = (b²-1)/2+b+(b²+1)/2 = (b²-1+2b+b²+1)/2 = (2b²+2b)/2 = b²+b
We check it with b=89, p=89²+89=7921+1=8010
We can deduce the following property...
For each prime number b, there exists a Pythagorean triplet (a, b, c) of non-zero natural integers satisfying the Pythagorean relation a²+b²=c² with c=(b²+1)/2 and a=(b² -1)/2
Your formula is great. If b=3 than c=5 and a=4 . Fits best !
"Also there is only one possible solution" FALSE. There is only one possible solution IN INTEGERS, but there are infinitely many non-integer solutions: a = 15, b = 89, c = sqrt(8146) = 90.255193756..., and P = 194.255193756... is a solution; a = 200, b = 89, c = sqrt(47921) = 218.9086567..., and P = 507.9086567... is another solution; etc.
@@douglasmiller1233
We are looking for sides belonging to Z+. In fact we are looking for a Pythagorean triple, and therefore only integers.
Your solution is only one of an infinite number of solutions. Side a could be 89, and we would have a right triangle with 2 45 degree angles. If I choose a value of 2 X 89 = 178 for c, then my right triangle would be a 30 60 90 right triangle. If you were one of my grade school math students, I would assign the following homework question for you: "How many angles and/or side lengths are required to uniquely specify any polygon ?"
0:56 “Keep in your mind that the side lengths must be a positive integer”, See diagram: Sides ∈ Z+
Nothing need be prime, and the given value can be an irrational square root (for example) and so long as the number whose square root is taken is factorable, you will have a solution for every possible combination of the factors (BASED ON the factors, not the factors directly). And the given one, of course, with the two unknown sides being a single unit apart (for purists who will be apoplectic realizing I mean "one times a number" to be considered a prime factorization). So if the known value is the square root of 255, 1*255, 3*85, 5*51, and 15*17 will all generate solutions.
(By the way, that last fact is why one uses a single pair of primes generating an encryption solution: using several gives the codebreaker several possible solutions.))
For example, from my last: 3*85. (85+3)/2 and (85-3)/2 are the two sides.
You don`t need to know what c and a equal. All you need is what c+a is equal to. You add b and you have the perimiter.
畢氏數(Pythagorean triple)有通解(General solutions) ：
(b,(b²-1/2),(b²+1)/2),當b為奇數(odd)，或(2b,b²-1,b²+1)
Surely there are many integer possibilities for a and c. You just need to push the point opposite the 89 length and a and c will change whilst 89 remains the same. I think this is a possible solution but not THE solution as it cannot be defined.
Mathematical 'magic' was used here, because, in fact, as long as you don't have one more side or one more angle (except of the right one, of course), you have an infinite number of solutions.
actually no, the prob says integer numbers on the sides, that narrow it down to only 1 solution.
0:56 “Keep in your mind that the side lengths must be a positive integer”, See diagram: Sides ∈ Z+
It looked like a 30, 60, 90 triangle so I took the given shortest side and formed three sides in the ratio of 1, 2, and root three. This produced sides of 89, 178, and 89 root 3. This checks with the Pythagorean Triple 7921 + 23763 = 31684.
When all are integers, a^2 = c^2 - b^2 c= (a^2 + 1)/2, b = c-1. Works for a=3, b=4, c=5; works for a=5, b=12, c=13. But it doesn't work for a=4. Works for a=21, b=220, c=221. I'm guessing it works for any a except if a itself is a square. Nope, a=9, b=40, c=41 works. I guess it works only when a is odd. Works for a=25, b=312, c=313.
Since the Triangle Inequality includes degenerate triangles it could be argued that a=0 does give an acceptable second answer for perimeter of 89+89+0=178.
Thank you!
Something does not make sense. Can you not move the point C to the right keeping given side length at fixed 89 and thus change the sum of other two sides? By moving the point c anywhere on the line you would still keep the side length 'b' constant at 89 but change the perimeter of the triangle.
Go to 1:00 it shows him making a circle
around the Z+. It means the side lengths can only be whole numbers.
When using the (89)(89) choice, the simultaneous equations can be solved in the same manner as with the (7921)(1) choice, namely by addition to eliminate "a".
Amazing 👍
Thanks for sharing 😊
Respected Sir 🙏, I like the way of your answering
Solutions may not be full filled
Pythagoras values
Please check this sir.
Surely there are many possible solutions
He said they are integers.
Since 89 is a prime number, there is only one solution 🧐
As they are positive integers
and the number(89 Square) is PRIME having only one solution
THERE IS ONLY ONE SOLUTION YOU FOOL
@@BruceArnold318 Ah, I missed that bit too. I was scratching my head thinking that the number of solutions is infinite.
There’s only one….and stop calling me Shirley😂
So, if you were asked to find the perimeter of the right triangle with limited information, like we have, you could do as the video did.
It finds one possible perimeter. But if there are other values to the base, different hypotenuses will also result, leading to different perimeters.
If the sides are integers (positive) there are many solutions. If the integer requirement is lifted, we have an infinite number of bases, hypotenuses and perimeters!
Perimeter is greater than 178
If "a" was zero then "c "would be 89.
Any value for "a" would increase "c"
So ....the perimeter is 89+ (>89) + (>0) or >178
I'm definitely coming back to this to give it a try.
89^2=(c-a)(c+a), 89 is prime, 89^2=1×89^2 89×89 89^2×1, thus c-a=1, c+a=89^2, 2a+1=89^2, a=3960, c=3961, therefore the perimeter is 89+3960+3961=8010😊
Excellent!
Thanks for sharing! Cheers!
You are awesome. Keep it up 👍
no need to solve for a,c values.. we already got c+a=89^2 ; we need perimeter (c+a)+b = 89^2+89 = 8010
If moves along the line BC, it’is obvious that the perimeter varies from zero to infinite. It should be specified that the solution is a set of Pythagorion numbers
0:56 “Keep in your mind that the side lengths must be a positive integer”, See diagram: Sides ∈ Z+
the title page is misleading because it fails to say that the sides are integers.
You forgot to mention your condition that only whole numbers apply. If not, 7921 can also be divided by any other number less than 7921 to produce a fraction, e.g. 7921=(100)(79.21).
In that case c=89.605 and a=10.395.
The circumference is then 189.
This problem therefore gives an infinite number of answers.
(You also don't have to calculate a and c separately. If you know that (a+c) is a value, you can add the known value b.)
Obviously, the larger value is more acceptable here.
Very easy
Doubt this, what will happen if on the drawing BC is reduced by8 units? You do not have th angles of the BAC and ACB?
You don't have enough info to calculate a and c . You either have to know 2 of the 3 sides or know the angle of one of the non right angle sides- you have neither. The side described would be a sliver and not look at all like the triangle drawn. So you can randomly find an infinite number of right triangles with one side of 89 units.
There are infinite solutions for this question due the given information.
That’s very nice
Thanks Sir
Thanks PreMath
❤❤❤❤❤
Always welcome
You are awesome. Keep it up 👍
At 1:00 of the video he does state the sides are positive integers. Otherwise it would be impossible to solve the problem. In the diagram it would have been better to state this in words rather than stating "sides E Z+". Also the the perimeter question is meaningless. It would have been better to just ask for the lengths of the other two sides.
He does say the side lengths must be a positive integer. Otherwise, there would be an infinite number of answers. Also, the diagram does show Sides E Z+ although that's a clumsy way to indicate the sides are positive numbers greater than 0. I would have written, "The sides are integers".
In the diagram: Sides ∈ Z+ is clumsy math talk for "sides are elements of the set of positive integers"
Elegant way of solving the problem, but can a hypotenuse of 3961 be correct? It doesn't seem reasonable. That would make angle A about 88.7 degrees. BTW, love your videos. I try to solve several each day. (with your wonderful help, of course).
As long as the angle A is less than 90°, we have a triangle, no matter how long is the hypotenuse 🤓
I'm no math guru but I it seems to me that there are infinite answers depending on the position of point "C" relative to "B".
the sides must be integer numbers..theres only 1 solution.
Probably could solve this much easier using trig to find side BC using arctan(). And then Pythagorean theorem to find side AC.
So...arctan(89/BC) =
If the angle at A changes without changing the length of AB, will the answer be the same?
There are an infinite number of solutions to this problem depending on the slope of the hypotenuse
Line BC=89 line AC= 89*2^(1/2) is also an answer!
If the hypotenuse of the right triangle is 89 what is the perimeter and it's area?
Yes but what if the hypotenuse is a gorilla. This is overlooked more often than we realize.
A triangle can NOT be described/defined by one angle and one side. The given answer is correct but is one of many.
I can not see the point of even attempting to solve it!!!
There is an important detail in the upper right corner of the video: _sides_ ∈ ℤ⁺ 🧐
Aren't there infinite perimeters?
Yes, there are infinite solutions to the perimeter because of lack of information.
@@ra15899550 theres only 1 solution, the prob says the sides must be integer numbers, i had the same concern but thats the correct answer.
U are all read the prob carefully, sides are real nos, always dont try to pick up mistakes only, u fit 4 only that, develop positive attitude first, give suggestions like me better
Mukundsir
Un triangolo rettangolo con i lati : a - b - c (ipotenusa !)
Conoscendo soltanto il valore di un solo lato a
a = 3-5-7-9-11-13-15-fino all’infinito !
Come calcolare i lati : b e
L’ ipotenusa : c ?
a = 5 ; b = 12 ; c = 13
Prova : 5^2+12^2 =13^2
25 +144 = 169
Come calcolare : b e c ?
Con a = 3-5-7-11-13 numero primo (una soluzione)
Con a = 9-15 ( multiplo di 3) almeno due soluzioni !
a=9 ; b=40 ; c=41
9^2 + 40^2 = 41^2
81 + 1600 = 1681
Altra soluzione :
a=9 ; b=12 ; c=15
9^2 + 12^2 =15^2
81 + 144 = 225
Pazzesco !
Con a = 33 (11x3)
Esistono…
4 soluzioni !
33^2+44^2 = 55^2
33^2+56^2 = 65^2
33^2+180^2=183^2
33^2+544^2=545^2
Potete spiegare perché ?
I'm not a 'smart' man, but as I don't see explicitly where the sides & perimeter have to be all INTEGERS, then I'm postulating this triangle to be isosceles with the perimeter being ~ 303.8650070512055
But what do I know? I probably missed something.
Figuring out c+a = 7921 is enough to answer the question. It is not necessary to add c+a and c-a. Just add 89 to 7921 and you find the answer. Why bother calculating c and a individually? Besides, this problem has multiple solutions unless the length of the known side is not a prime number, and infinite solutions if c and/or a are not integers.
yhea but the problem says integer numbers...so...
I loved this question!
❤️
Thanks for your feedback! Cheers! 😀
You are awesome. Keep it up 👍
AC can be any value that is greater than or equal to AB! Why did you assume that BC could not be zero? BC can be any postive value from 0 to infinity!
Why do you assume (a+c) and (a-c) are integers?
because a must be a integer and also c, so....
Okay
Very thanks
Seems that this is 'A" solution but not 'THE' solution because there are infinite valid solutions based on the scant data provided.
Am I missing something hare?
C'est une possibilité, ça pourrait aussi être une infinité d'autres solutions, non ?
Comme le plus petit des trois nombres est premier, il n'y a qu'un seul triplet pythagoricien possible ! 😉
@@ybodoN admettons pour l'exemple avec un triangle particulier, mais je ne vois pas ce qui empêche d'avoir la base et l'hypoténuse de longueur quelconque
@@olivierjosephdeloris8153 on a un angle droit et les trois côtés doivent correspondre à des entiers naturels, ce qui implique un triplet pythagoricien.
Quand le plus petit des trois nombres est impair, une des solutions est (n, m, m + 1) où m + 1 = ½ (n² + 1). Quand n est premier, c'est la seule solution.
@@ybodoNd'accord, en effet la contrainte des nombres entiers, ça change tout. Le Z+ m'avait échappé
thank you
Similar using for side b à prime number, à good idea for fun.
Well, that is one answer of many.
Will this solution satisfy Pythagorean solution.
Yes.
3961^2 - 3960^2
factors as
(3961- 3960) * (3961+3960) = 1 * 89^2.
Google "Euclid's gnomon".
No need to find each side separately
We know that one of the sides is 89 and the sum of the other two sides is 89^2
P=89, b=3960, h=3961, May be
So you just guessed it really! You have not actually found an answer just two whole numbers that fit Pythagorean theorem.
Equally a could be 89 and so c 125.8.
Would have been easier to plug into c-a=1, so a=c-1
Thanks for your feedback! Cheers! 😀
I am confused
Assume side a =1 then side c = square root of 7922 this gives a different Perimeter
Assume side a =1oo then side c = square root of 17922 this gives a different Perimeter
0:56 “Keep in your mind that the side lengths must be a positive integer”, See diagram: Sides ∈ Z+
Il y a une infinité de valeurs de a et c, ainsi pour le périmètre
I thought so too but he said they are integers.
Comme le plus petit des trois nombres est premier, il n'y a qu'un seul triplet pythagoricien possible ! 😉
@@BruceArnold318merci, je ne suis pas très bon en anglais, je n'avais pas saisi
@@ybodoNje ne comprend pas vraiment bien l'anglais et je n'avais saisi : appartient à Z. Merci pour votre réponse
Je reste d'accord avec vous: il y a une infinité de solutions.
This is just another one of those mathematical exercises that serve only as a mathematical curiosity but without any practical use. like something that exists just to make teachers horny in the classroom but we will never see an engineer having to solve a similar problem in their work.
So math lessons should be limited to what an engineer might see? What if he has poor eyesight? Mr. Magoo's Math 😂
Et si on augmente l'angle BAC ?
A sera toujours de 89 mais les deux autres côtés auront augmenté...
The Perimeter is any value that is equal to or greater than 89!
But c^2=a^2 + b^2
Does not add up
What is the 5 term of sequence given -8,-3,2,7,_,_,22
12
There can be numerous triangles with this information??? AC side can be anything more than 89? No?
Wow! I used trigonometry, but that got me nowhere.
Great solution!
anybody can see that given only the length of one side the problem has infinite solutions
Assume the lengths are positive integers.
0:56 “Keep in your mind that the side lengths must be a positive integer”, See diagram: Sides ∈ Z+
As long as you eliminate a=0, P=89+a+c=89+7921 didn't really need to solve for the 2 sides.
But side “a” couldn’t be equal to zero! It must have a value other than zero. If side “a” is zero, then the figure could not anymore be a triangle!
@@dawon7750pls watch video before commenting. 7:00
There are many solutions
The only limit is the possible maximum length that side c can take to remain a orthogonal triangle
So my friend it seems to me you are out fishing
Something is wrong with your geometry
In real the problem has infinity answers
Once again you assume interger values for the sides. If you take as a guess one of the sides is length 1 you will NOT get you calculated value of the perimeter. You are doing a disservice to mathematics by posting these solutions as it appears to the unsuspecting that this is the only possible result!
Baba, you should tell the angle then only one solution will emerge
Simple steps... but a magnificent sequence!
89 is prime number given, so the solution became possible
Sorry, but it seems a little convincing "solution"
a = 8, b = 15, c = 17 perimeter = 40; OR a = 8, b = 6, c = 10 perimeter = 24. This proves the fallacy of this video.
You’re quite mistaken. What you’ve shown is that there are some whole numbers (such as 8) that belong to more than one Pythagorean triple. But that doesn’t mean it’s true of EVERY whole number. Some whole numbers (including all odd primes, such as 89) belong to only one such triple.
There is not enough information given to solve this one.
this solution is wrong. as a2+b2is not equal to c2. (3960)2 + (3960)2 is not and never be equal to (89)2. such a triangle is not possible
who says the sides have to be whole numbers?
emmm..the problem?
Excuse my rudness --= without additional information it is not possible to derive ANY answer. It all depends on knowing one more side or one more angle.
C can be ANYWHERE--thus there is nomanswer possible with ithe info given. Total BS
He threw in an extra requirement that a and c differed by only1. That's cheating. Bad problem.
The 3rd Binomial formula was the key, very good.
Wrong. The solution is infinite
You need another data point to make a finite solution problem
this is bogus. There are an infinite number of solutions unless you know 2 sides or 2 angles. Imagine an 89 inch line that intersects an infinite line line at a right angle. From the top of that line, you can place another line down to the long line at any angle > 0 and < 90 so that the circumference is 178.00001 to just under infinity
actually theres only one whit the given information...is not that hard to see, 178.00001 is not an integer so...isnot a solution to the problem.
一個方程式（畢氏定理）兩個未知數，故有無窮盡的解。需再加一條件，例如邊長是整數方可解出另二邊長，此視頻就是這樣設定的。
Um. You'd better rethink that. You do not have enough information to solve it.
208 ÷2
That was amazing. I wouldn't have thought you could do it with TWO missing triangle sides. One, yes. But not two. Well, I stand corrected. Very memorable.
Only because it's in Z+
In R there is an infinity of solutions.
@@esunisen3862
I have no idea whatsoever as to what you just said. Thanks?
@@sail2byzantium Esunisen was trying to tell you Z+ is the little twist
PreMath put on this puzzle. Go to 1:00 it shows him making a circle
around the Z+. It means the side lengths can only be whole numbers.
The clickbait picture only shows a right triangle with on side being 89.
The true answers are infinite. You CAN NOT solve a triangle with only 2 pieces of information. One known angle of 90 deg. & one side measurement of 89 is not enough information. You always need at least 3 pieces of info to solve. PreMath gives the 3rd piece as (each side must be a whole number).
It is kinda trichery, but it is a good math lesson..
@@simpleman283 Thanks for clarifying this point! I thought there surely can be no unique solution with only two pieces of information, but was then befuddled when he managed to produce an answer. Trickery, indeed! :)