**CBSE NCERT Solutions Chapter 6 Triangles Exercise 6.4 Question 7**

**7.** Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

**Solution: **

**Given: **ABCD is a square, AEB is an equilateral triangle described on the side of the square, DBF is an equilateral triangle described on diagonal BD of square.

**To Prove: **

**Proof: **Any two equilateral triangles are similar because all angles are of 60 degrees.

Therefore, by **AAA similarity criterion**, ~

**(1)**

** {The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides}**

But, we have {Diagonal of square is times of its side} **(2)**

Putting equation** (2)** in equation **(1), **we get

= 2

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

**Hence Proved**