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 ……..

 

Let side of triangle be 'a'=2√2. Altitude H = a√3/2 = √6.
Let rectangle have dimensions l, w.

Let base l be on the base of triangle. w is the height.
Triangle above rectangle is similar to large triangle.
(H-w)/H = l/a.
l = a (H-w)/H = a (1-w/H).
Area A = lw = aw (1-w/H).
A' (w) = a (1-2w/H) = 0 ⇒ w = H/2 = √6/2.
l = a/2 = √2.
Max Area A = √2 * √6/2 = √3.
Square of area = 3.

<div class="photo-widget-full"><div class="figure"><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1755674293phpw13VFq.jpeg" alt="" width="207" height="208" loading="lazy"></picture></div><div class="figure">Let side of triangle be 'a'=2√2. Altitude H = a√3/2 = √6.<br>Let rectangle have dimensions l, w.</div><div class="figure">&lt;!- [if !supportLineBreakNewLine]-&gt;&lt;!- [endif]-&gt;Let base l be on the base of triangle. w is the height.<br>Triangle above rectangle is similar to large triangle.<br> (H-w)/H = l/a.<br>l = a (H-w)/H = a (1-w/H).<br>Area A = lw = aw (1-w/H).<br>A' (w) = a (1-2w/H) = 0 ⇒ w = H/2 = √6/2.<br>l = a/2 = √2.<br>Max Area A = √2 * √6/2 = √3.<br>Square of area = 3.</div></div>

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