The hypotenuse of an isosceles right-angled triangle is q. If we describe equilateral triangles (outwards) on all its three sides, -Maths 9th

The hypotenuse of an isosceles right-angled triangle is q. If we describe equilateral triangles (outwards) on all its three sides, -Maths 9th
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Answer :

(b)  \(rac{q^2}{4}\) (2√3 + 1).AC = q, ∠ABC = 90º ⇒ q = \(\sqrt{AB^2+BC^2}\)⇒ q = \(\sqrt{2x^2}\)⇒ q2 = 2x2 ⇒ \(x\) = \(rac{q}{\sqrt2}\)∴ Area of the re-entrant hexagon = Sum of areas of (ΔABC + ΔADC + ΔBFC + ΔAEB) = \(rac{1}{2}\)x \(rac{q}{\sqrt2}\) x \(rac{q}{\sqrt2}\) + \(rac{\sqrt3}{4}\)q2 + \(rac{\sqrt3}{4}\)\(\big(rac{q}{\sqrt2}\big)^2\) + \(rac{\sqrt3}{4}\)\(\big(rac{q}{\sqrt2}\big)^2\)= \(rac{q^2}{4}\) + \(rac{\sqrt3}{4}\)q2 + \(rac{\sqrt3}{8}\)q2 + \(rac{\sqrt3q^2}{8}\) = \(rac{q^2}{4}\) (2√3 + 1).
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