If a rectangle is inscribed in an equilateral triangle of side length 2 2 as shown in the figure, then the square of the largest area of such a rectangle is _____.

Question

If a rectangle is inscribed in an equilateral triangle of side length 2 2 as shown in the figure, then the square of the largest area of such a rectangle is _____.,

MARKS App · Accepted Answer

In Δ D B F , tan 60 ° = 2 b 2 2 - ℓ ⇒ b = 3 2 2 - ℓ 2 A = Area of rectangle = ℓ × b ⇒ A = ℓ × 3 2 2 2 - ℓ On differentiating, we get d A d ℓ = 3 2 2 2 - ℓ - ℓ · 3 2 = 0 ⇒ ℓ = 2 Now, A = ℓ × b = 2 × 3 2 2 = 3 ⇒ A 2 = 3 .

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