Question

Find the length of the hypotenuse of an isosceles right-angled triangle whose area is 200 cm². Also, find its perimeter. Take $\sqrt{2}$ = 1.414.

Answer 1

In a right isosceles triangle, $base=height=a$

Therefore,
$Areaofthetriangle=\frac{1}{2}\times base\times height=\frac{1}{2}\times a\times a=\frac{1}{2}{a}^2$

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 = $\sqrt{a^2+a^2}$

$$\begin{gathered} =\sqrt{2a^2} \\ =a\sqrt{2} \\ =20\times 1.414 \\ =28.28\mathrm{cm} \end{gathered}$$

Perimeter of the triangle = $a+a+c$
​ $=20+20+28.28$

= 68.28 cm

The length of the hypotenuse is 28.28 cm and the perimeter of the triangle is 68.28 cm.