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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)

ΔAPCΔBQC [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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