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Question

# Find the length of the hypotenuse of an isosceles right-angled triangle whose area is 200 cm2. Also, find its perimeter. [Given: $\sqrt{2}$ = 1.41]

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Solution

## In a right isosceles triangle, $\mathrm{base}=\mathrm{height}=a$ Therefore, $\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{triangle}=\frac{1}{2}×\mathrm{base}×\mathrm{height}=\frac{1}{2}×a×a=\frac{1}{2}{a}^{2}\phantom{\rule{0ex}{0ex}}$ Further, given that area of isosceles right triangle = 200 cm2 $⇒\frac{1}{2}{a}^{2}=200\phantom{\rule{0ex}{0ex}}⇒{a}^{2}=400\phantom{\rule{0ex}{0ex}}or,a=\sqrt{400}=20\mathrm{cm}$ In an isosceles right triangle, two sides are equal ('a') and the third side is the hypotenuse, i.e. 'c' Therefore, c = $\sqrt{{a}^{2}+{a}^{2}}$ $=\sqrt{2{a}^{2}}\phantom{\rule{0ex}{0ex}}=a\sqrt{2}\phantom{\rule{0ex}{0ex}}=20×1.41\phantom{\rule{0ex}{0ex}}=28.2\mathrm{cm}$ Perimeter of the triangle = $a+a+c$ ​ $=20+20+28.2$ = 68.2 cm The length of the hypotenuse is 28.2 cm and the perimeter of the triangle is 68.2 cm.

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