Question

the length of a rectangular and is twice its lateral triangle are song on its length and is diagonal prove that the ratio of their area is 1 ratio 4 ratio 5​

Answer 1

Answer:

Here ABCD is a square, AEB is an equilateral triangle described on the side of the square and DBF is an equilateral triangle described on diagonal BD of the square.

To prove: Ar.(△DBF/△AEB)=2/1

It two equilateral triangles are similar, then all angles are =60 degrees.

Therefore, by AAA similarity criterion, △DBF∼△AEB

Ar.△AEBAr.△DBF=AB2DB2    .....(1)

We know that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

But, we have DB=2AB    .....(But diagonal of square is 2 times of its side)     ....(2)

Substitute equation (2) in equation (1), we get

Ar.△AEBAr.△DBF=AB2(2AB)2=2

Therefore, area of equilateral triangle described on one side of square is equal to half the area of the equilateral triangle described on one of its diagonals.