Question

# Question 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 whose one diagonal is AC. ΔAPC and ΔBQC are two equilateral triangles described on the diagonal AC and side BC of the square ABCD. To Prove that area(ΔBQC)=12area(ΔAPC) Proof  ΔAPC and ΔBQC are both equilateral triangles (Given) ∴ΔAPC∼ΔBQC [AAA similarity criterion] ∴area(ΔAPC)area(ΔBQC)=AC2BC2 ⇒(√2BCBC)2=2BC2BC2=2[Since,Diagonal=√2side=√2BC] ⇒area(ΔAPC)=2×area(ΔBQC) ⇒area(ΔBQC)=12area(ΔAPC)

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