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QJEEM21Ma358c: JEE Main 2021 Mathematics - Class 11 JEEM21Ma358c Solution: If a rectangle is inscribed in an equilateral triangle of side…

If a rectangle is inscribed in an equilateral triangle of side length 222\sqrt 2 as shown in the figure, then the square of the largest area of such a rectangle is _____________.

JEE Main 2021 (Online) 25th July Evening Shift Mathematics - Properties of Triangle Question 15 English

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JEE Main 2021 (Online) 25th July Evening Shift Mathematics - Properties of Triangle Question 15 English Explanation

In Δ\Delta DBF

tan⁡60∘=2b22−l⇒b=3(22−l)2\tan 60^\circ = {{2b} \over {2\sqrt 2 - l}} \Rightarrow b = {{\sqrt 3 \left( {2\sqrt 2 - l} \right)} \over 2}

A = Area of rectangle = l ×\times b

A=l×32(22−l)A = l \times {{\sqrt 3 } \over 2}\left( {2\sqrt 2 - l} \right)

dAdl=32(22−l)−l.32=0{{dA} \over {dl}} = {{\sqrt 3 } \over 2}\left( {2\sqrt 2 - l} \right) - {{l.\sqrt 3 } \over 2} = 0

l=2l = \sqrt 2

A=l×b=2×32(2)=3A = l \times b = \sqrt 2 \times {{\sqrt 3 } \over 2}\left( {\sqrt 2 } \right) = \sqrt 3

⇒\Rightarrow A2 = 3

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