Question

If the area of an equilateral triangle described on diagonal of a square is 84 sq. cm then the area of an equilateral triangle described on the side of the same square is

a. 44 sq.cm

b. 42 sq. cm

c. 11 sq cm​​

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.△AEB

Ar.△DBF

=

AB

2

DB

2

.....(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=

2

AB .....(But diagonal of square is

2

times of its side) ....(2)

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

Ar.△AEB

Ar.△DBF

=

AB

2

(

2

AB)

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.