Finding the Hypotenuse of an Isosceles Right Triangle

We are given an isosceles right-angled triangle with an area of $81 \text{ cm}^2$. We need to find the length of its hypotenuse.

Area Calculation for Isosceles Right Triangle

Let the length of the two equal legs of the isosceles right-angled triangle be '$a$' and the length of the hypotenuse be '$h$'.

The area of a triangle is given by the formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$

In an isosceles right-angled triangle, the two legs are perpendicular. So, we can set base = $a$ and height = $a$. Area = $\frac{1}{2} \times a \times a = \frac{a^2}{2}$

Using the Pythagorean theorem, $h^2 = a^2 + a^2 = 2a^2$. This means $a^2 = \frac{h^2}{2}$.

Substituting $a^2$ in the area formula:

Area = $\frac{1}{2} \times \frac{h^2}{2} = \frac{h^2}{4}$

Calculating the Hypotenuse Length

We are given that the Area = $81 \text{ cm}^2$. Using the derived formula:

$\frac{h^2}{4} = 81 \text{ cm}^2$

Now, solve for $h^2$: $h^2 = 81 \times 4$ $h^2 = 324$

To find the hypotenuse $h$, take the square root of both sides:

$h = \sqrt{324}$ $h = 18$ cm

Therefore, the length of the hypotenuse is 18 cm.