The area of either rectangle is (7)(21) = 147. Using the value of x that PreMath computed, each white triangle has area (1/2)(b)(h) = (1/2)(x)(7) = (1/2)(196/21)(7) = 196/6. There are 2 of these triangles inside either rectangle, combined triangle area equals 196/3. So take the area of either rectangle and deduct the combined area of 2 triangles to get the purple area: 147 - 196/3 = 441/3 - 196/3 = 245/3 cm² or 81.67 cm², as PreMath also found.
Drawing diagonal. Tan of half the parallelogram acute angle = 7/21= 0.333. Tan-1 = 18.435 degrees. Doubling this = 36.87 degrees. In white triangle, tan 36.87 = 7/ base length. Base length = 7/ tan 36.87 = 9.333. Base length of parallelogram = 21 - 9.333 = 11.667. Area = 11.667 x 7. 81.67.
Draw the diagonal d and if h is its perpendicular bisector from the intersection, then by Similarity, h/(d/2) = 7/21 h = d/6 Purple Area = 2(½*d*h) = d²/6 We know d² = 7² + 21² = 490 Purple Area = 245/3
Professor sir, After having found out the value of ' X ' , we could arrive at the area of the shaded portion by simply deducting the area of 2 small triangles on the left and right from area of the rectangle. That is 21x7= 147. 147 - 65.33 = 81.67
Or we could have set the hypotenuse of the triangle to x and the longer leg to 21-x instead, and once we found x (35/3) we could simply calculate area of parallelogram as height * base = 7x = 245/3 = 81.67
By observation, the four white triangles are all similar by AA similarity. Each, being formed from the corner of a rectangle, is a right triangle, and each shares a vertical angle with the one to its left or right. By the fact that opposite sides of a rectangle are parallel, those two pairs of vertical angles are identical. Investigating the dimensions of the triangles, it can be seen that short leg of each triangle is 7, the short side of the rectangle, and that the long leg and hypotenuse add up to 21, the long side of the rectangle. Let x be the length of the long leg of the triangle: White triangle: 7² + x² = (21-x)² 49 + x² = 441 - 42x + x² 42x = 441 - 49 = 392 x = 392/42 = 28/3 The purple area is equal to one of the rectangles minus two of the triangles. A = hw - 2(bh/2) A = 7(21) - (28/3)7 A = 147 - 196/3 A = 441/3 - 196/3 A = 245/3 ≈ 81.67 cm²
You need to simplify: x= 196/21 = 28/3 Also, there is an easier way to solve for x: Let: y = hypotenuse of the green triangle We have: (Eq 1) x+y = 21 (Eq 2) 7^2 + x^2 = y^2 (pythagorean theorem) -> (x-y)(x+y)=-7^2 (by Eq 2) x-y = -7^2 / 21 = -7/3 (by Eq 1) x = (21 -7/3)/2 = 28/3 (by Eq 1)
The other rectangle is made by rotating about the diagonal. Generally the rectangle is a × b, x^2=a^2+(b-x)^2=a^2+b^2+x^2-2bx, x=(a^2+b^2)/2b, the area is a(a^2+b^2)/2b.😊
La zona sombreada o solapo de los dos rectángulos es un rombo cuya superficie es la mitad del producto de sus dos diagonales, D y d → D²=7²+21²=490 → D=√490 →→ El triángulo rectángulo de lados 7/21/D es semejante al de lados b/7/d → Razón de semejanza s=7/21→ d=s*D → Área zona sombreada =D*d/2 =D*s*D/2 =s*D²/2 =(7/21)(√490)²/2 =7*490/21*2=81.6666 Gracias y saludos.
The length of each side of the purple area is x x^2 = 7^2 + (21 - x)^ 2 x^2 = 7^2 + 21^2 - 2(21)x + x^ 2 0 = 7^2 + 21^2 - 2(21)x 2(21)x = 7^2 + 21^2 2(3)x = 7 + 3(21) = 70 3x = 35 x = 35/3 The area of the purple area = 7 (35/3) = 245/3
Hi everyone. Let us consider the largest diagonal of rhombus to be the diagonal of the rectangle given by √(7²+21²)=7√10 which is equal to Smallest side of rectangle ÷ sin θ Hence sinθ = 1/√10 Again the smallest diagonal of rhombus is given by, Smallest side of the rectangle ÷ cos θ Where cosθ = √(1-sin²θ) =3/√10 Therefore the area of rhombus = largest diagonal × 1/2 of smallest diagonal = 7√10 × (1/2) × (7√10)/3 = 490/6= 81.67 square units.
@ 4:59 , ...🤔 I know main properties of multiplication of integers as Closure Commutative Associative Distributive Identity and Multiplication by Zero. ...but that "New Math of Criss Cross always throws me for a curve. Cross Multiplication, yes...Criss Cross ...not so much. 🙂 Now the Butterfly Effect, ...that I can relate too. 😉
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long diagonal:
d² = 7² + 21² = 49 + 441 = 490
d = √490 = 7√10
short diagonal:
e = 7√10 *7/21
e = 7√10 / 3
area of deltoid:
A = 1/2 * 7√10 * 7√10/3
A = 1/6 * 49 * 10
A = 490/6 = 245/3 ≈ 81.67 cm²
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Great puzzle!
The area of either rectangle is (7)(21) = 147. Using the value of x that PreMath computed, each white triangle has area (1/2)(b)(h) = (1/2)(x)(7) = (1/2)(196/21)(7) = 196/6. There are 2 of these triangles inside either rectangle, combined triangle area equals 196/3. So take the area of either rectangle and deduct the combined area of 2 triangles to get the purple area: 147 - 196/3 = 441/3 - 196/3 = 245/3 cm² or 81.67 cm², as PreMath also found.
Drawing diagonal.
Tan of half the parallelogram acute angle = 7/21= 0.333.
Tan-1 = 18.435 degrees.
Doubling this = 36.87 degrees.
In white triangle, tan 36.87 = 7/ base length.
Base length = 7/ tan 36.87 = 9.333.
Base length of parallelogram = 21 - 9.333 = 11.667.
Area = 11.667 x 7.
81.67.
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Amazing 👍
Thanks for sharing😊
Thanks Sir
Thanks PreMath
That’s nice
Draw the diagonal d and if h is its perpendicular bisector from the intersection, then by Similarity,
h/(d/2) = 7/21
h = d/6
Purple Area = 2(½*d*h) = d²/6
We know d² = 7² + 21² = 490
Purple Area = 245/3
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Generalized: the area of the purple rhombus is _½ (a² + b²) a/b_ where _a_ is the width of the rectangle and _b_ is its length.
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Sir, I think here it should be 1/6 in place of 1/2
Required area = 1/6 (a² + b²)
@@D.P9 then the formula would only work when the width of the rectangle is one third of its length 🤔
tan α = 7/21 = 1/3
α = 18,435°
sin 2α = 7 / b
b = 7 / sin 2α
b = 11,667 cm
Area = b . h
Area = 11,667 . 7
Area = 81,667 cm² ( Solved √ )
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Professor sir,
After having found out the value of ' X ' , we could arrive at the area of the shaded portion by simply deducting the area of 2 small triangles on the left and right from area of the rectangle.
That is 21x7= 147.
147 - 65.33 = 81.67
Excellent!
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You are awesome. Keep it up 👍
Or we could have set the hypotenuse of the triangle to x and the longer leg to 21-x instead, and once we found x (35/3) we could simply calculate area of parallelogram as height * base = 7x = 245/3 = 81.67
By observation, the four white triangles are all similar by AA similarity. Each, being formed from the corner of a rectangle, is a right triangle, and each shares a vertical angle with the one to its left or right. By the fact that opposite sides of a rectangle are parallel, those two pairs of vertical angles are identical. Investigating the dimensions of the triangles, it can be seen that short leg of each triangle is 7, the short side of the rectangle, and that the long leg and hypotenuse add up to 21, the long side of the rectangle. Let x be the length of the long leg of the triangle:
White triangle:
7² + x² = (21-x)²
49 + x² = 441 - 42x + x²
42x = 441 - 49 = 392
x = 392/42 = 28/3
The purple area is equal to one of the rectangles minus two of the triangles.
A = hw - 2(bh/2)
A = 7(21) - (28/3)7
A = 147 - 196/3
A = 441/3 - 196/3
A = 245/3 ≈ 81.67 cm²
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You need to simplify:
x= 196/21 = 28/3
Also, there is an easier way to solve for x:
Let: y = hypotenuse of the green triangle
We have:
(Eq 1) x+y = 21
(Eq 2) 7^2 + x^2 = y^2 (pythagorean theorem)
->
(x-y)(x+y)=-7^2 (by Eq 2)
x-y = -7^2 / 21 = -7/3 (by Eq 1)
x = (21 -7/3)/2 = 28/3 (by Eq 1)
Very good eh
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Thanks for video.Good luck sir!!!!!!!!!!!!!!!!!!!!!!!!
The other rectangle is made by rotating about the diagonal. Generally the rectangle is a × b, x^2=a^2+(b-x)^2=a^2+b^2+x^2-2bx, x=(a^2+b^2)/2b, the area is a(a^2+b^2)/2b.😊
نحسب القطر الكبير للشكل الأحمر 22.13 نقسم الشكل الأحمر الى أربع مثلثاث..ونحسب زاوية ميل القطر 18.43درجة..ثم نحسب مساحة المثلث الواحد ونضرب بأربعة
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@@PreMath Thank you
Thanxs sir ❤❤😊
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If the sides of a right triangle can be written in the form a, x and b-x, where a and b are known, then x can be found.
Yay! I solved the problem.
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La zona sombreada o solapo de los dos rectángulos es un rombo cuya superficie es la mitad del producto de sus dos diagonales, D y d → D²=7²+21²=490 → D=√490 →→ El triángulo rectángulo de lados 7/21/D es semejante al de lados b/7/d → Razón de semejanza s=7/21→ d=s*D → Área zona sombreada =D*d/2 =D*s*D/2 =s*D²/2 =(7/21)(√490)²/2 =7*490/21*2=81.6666
Gracias y saludos.
The length of each side of the purple area is x
x^2 = 7^2 + (21 - x)^ 2
x^2 = 7^2 + 21^2 - 2(21)x + x^ 2
0 = 7^2 + 21^2 - 2(21)x
2(21)x = 7^2 + 21^2
2(3)x = 7 + 3(21) = 70
3x = 35
x = 35/3
The area of the purple area = 7 (35/3) = 245/3
(0,0) (21,7) (35/3,0) twice the area of the triangle formed these three points S=|(21,7)x(35/3,0)|=245/3
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On peut aussi utiliser les intégrales !!!
Why you didn't devide 196/21 by 7? => 28/3
Is it simpler to find area 7*21-7*28/3?
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Can we not congruent both Upper small triangles by ASA criteria as one side and corresponding angles are equal ? Then value of x will 21/2
Many approaches are possible to find the solution to this problem!
Cheers
7/x=7/y, so y^2=x^2=(21-x)^2-7^2=21^2-7^2-42x, 0:09 x=(21^2-7^2)/42=28/3, y=21-28/3=35/3, therefore the answer is 35/3x7=245/3.
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Hi everyone.
Let us consider the largest diagonal of rhombus to be the diagonal of the rectangle given by
√(7²+21²)=7√10
which is equal to
Smallest side of rectangle ÷ sin θ
Hence sinθ = 1/√10
Again the smallest diagonal of rhombus is given by,
Smallest side of the rectangle ÷ cos θ
Where cosθ = √(1-sin²θ)
=3/√10
Therefore the area of rhombus = largest diagonal × 1/2 of smallest diagonal = 7√10 × (1/2) × (7√10)/3 = 490/6= 81.67 square units.
Nandri sir
மிகவும் வரவேற்கிறோம், அன்பே❤️
Damn nice to see Tamil comments !
21*7-7*7/TAN(2*TAN-1(7/21))=81.67 Tangente is always your friend
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@ 4:59 , ...🤔 I know main properties of multiplication of integers as Closure Commutative Associative Distributive Identity and Multiplication by Zero. ...but that "New Math of Criss Cross always throws me for a curve.
Cross Multiplication, yes...Criss Cross ...not so much. 🙂
Now the Butterfly Effect, ...that I can relate too. 😉
Thanks for your feedback! Cheers! 😀
You are awesome. Keep rocking 👍
Gostei muito, obrigada
De nada ❤️
Obrigado pelo seu feedback! Saúde! 😀
Você é incrível. Continue assim 👍
28/3 is the lowest term of 196/21
245/3 considering that all white triangles are congruents
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I decided the same
A= 81,67 cm^2
You labeled 2 segments x without saying why they are equal.
245/3
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81.66 I'll say.
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You lost me at 1:30 when symmetry appeared with no explanation.
You're normally so detailed in your explanations, but this seems quite a jump.
We have identical rectangles😀