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]

Answer 1

In a right isosceles triangle, $\mathrm{base}=\mathrm{height}=a$

Therefore,
$\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{triangle}=\frac{1}{2}\times \mathrm{base}\times \mathrm{height}=\frac{1}{2}\times a\times a=\frac{1}{2}{a}^2$

Further, given that area of isosceles right triangle = 200 cm²

$$\begin{gathered} \Rightarrow \frac{1}{2}{a}^2=200 \\ \Rightarrow a^2=400 \\ or,a=\sqrt{400}=20\mathrm{cm} \end{gathered}$$

In an isosceles right triangle, two sides are equal ('a') and the third side is the hypotenuse, i.e. 'c'

Therefore, c = $\sqrt{a^2+a^2}$

$$\begin{gathered} =\sqrt{2a^2} \\ =a\sqrt{2} \\ =20\times 1.41 \\ =28.2\mathrm{cm} \end{gathered}$$

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.