Find the length of hypotenuse of an isosceles right angled triangle, having an area of 200 cm².

Let ABC be an isosceles right-angled triangle.

Area = 200 cm²

Let, side AB = BC = x cm and hypotenuse AC = h cm

We know that,

$\Rightarrow \text{Area of triangle } = \dfrac{1}{2} \times base \times height \\[1em] \Rightarrow \text{Area of triangle } = \dfrac{1}{2} \times BC \times AB \\[1em] \Rightarrow \text{Area of triangle } = \dfrac{1}{2} \times x \times x \\[1em] \Rightarrow 200 = \dfrac{1}{2} \times x^2 \\[1em] \Rightarrow x^2 = 400 \\[1em] \Rightarrow x = \sqrt{400} \\[1em] \Rightarrow x = 20 \text{ cm}.$

Now, using the Pythagoras Theorem for the △ABC

⇒ AC² = AB² + BC²

⇒ AC² = (20)² + (20)²

⇒ AC² = 400 + 400

⇒ AC² = 800

⇒ AC = $\sqrt{800}$

⇒ AC = $\sqrt{400 × 2}$

⇒ AC = $20\sqrt{2}$

⇒ AC = 20 × 1.414

⇒ AC = 28.28 cm.

Hence, length of hypotenuse = 28.28 cm.