Prove that the area of an equilateral triangle described on one side of a square is equal to half th

For Competitive Exams

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 diagonals AC and side BC of the square ABCD.

Area(ΔBQC) = $\dfrac{1}{2}$ Area(ΔAPC)

Since, ΔAPC and ΔBQC are both equilateral triangles, as per given,

ΔAPC ~ ΔBQC [AAA similarity criterion]

$\dfrac{area(ΔAPC)}{area(ΔBQC)} = \dfrac{AC^2}{BC^2} = \dfrac{AC^2}{BC^2}$

Since, Diagonal = $\sqrt{2} side = \sqrt{2}$ BC = AC

area(ΔAPC) = 2 × area(ΔBQC)

area(ΔBQC) = $\dfrac{1}{2}$ area(ΔAPC)

Hence, proved

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