Question

the length of a rectangle is twice its breadth. equilateral triangles are drawn on its breadth , length and diagonal prove ratio of their area is 1:4:5​

Answer 1

Given:

The length of a rectangle is twice its breadth.

To Prove:

ratio of area of the obtained triangle is 1:4:5​

Solution:

Let the breadth of the rectangle is B

Then, As per the given condition

The Length of the rectangle is 2B

From the property of the rectangle the diagonal of the rectangle can be obtained by:

D² = L² + B²

• D² = (2B)² + B²
• D² = 4B² + B²
• D² = 5B²
• D = √5B

As we have given in the question that three equilateral triangles are drawn by taking the base as the length, breadth and diagonal of the given rectangle.

So the sides of the triangle are given by:

• S = B
• S' = 2B
• S'' = √5B

And the ratio of their areas is given by the square of the given sides because the common term will eliminate when you take the ratio of that

• A : A' : A" = S² : S'² : S''²
• A : A' : A" = B² : (2B)² : (√5B)²
• A : A' : A" = B² : 4B² : 5B²
• A : A' : A"  = 1 : 4 : 5

so, the ratio of area of the obtained triangle is 1:4:5​

hence proved