Finding the Hypotenuse of an Isosceles Right Triangle

This question asks us to find the length of the hypotenuse of an isosceles right-angled triangle given its area. An isosceles right-angled triangle is a special type of right-angled triangle where the two perpendicular sides (legs) are equal in length. These equal sides form the right angle.

Understanding the Isosceles Right Triangle and Area

In a right-angled triangle, the area is calculated using the formula:

Area $= \frac{1}{2} \times \text{base} \times \text{height}$

In an isosceles right triangle, the two equal sides act as the base and the height. Let's denote the length of each of the equal sides by 's'.

So, the area of the isosceles right triangle is:

Area $= \frac{1}{2} \times s \times s = \frac{1}{2}s^2$

Calculating the Length of the Equal Sides

We are given that the area of the triangle is 121 cm$^2$. Using the area formula, we can write:

$\frac{1}{2}s^2 = 121 \text{ cm}^2$

To find $s^2$, we multiply both sides by 2:

$s^2 = 121 \times 2$

$s^2 = 242$

The length of the equal side 's' is the square root of 242:

$s = \sqrt{242}$ cm

We can simplify $\sqrt{242}$ as $\sqrt{121 \times 2} = \sqrt{121} \times \sqrt{2} = 11\sqrt{2}$ cm. However, we don't necessarily need to find the value of 's' itself, as we will need $s^2$ for the next step.

Finding the Hypotenuse using the Pythagorean Theorem

In any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is known as the Pythagorean theorem.

Let 'h' be the length of the hypotenuse. The other two sides are the equal sides, each of length 's'. According to the Pythagorean theorem:

Hypotenuse$^2$ $=$ Side1$^2$ $+$ Side2$^2$

$h^2 = s^2 + s^2$

$h^2 = 2s^2$

We already found that $s^2 = 242$. Now we substitute this value into the equation for $h^2$:

$h^2 = 2 \times 242$

$h^2 = 484$

To find the length of the hypotenuse 'h', we take the square root of 484:

$h = \sqrt{484}$

We need to find the number whose square is 484. We can test some values:

• $20^2 = 400$
• $21^2 = 441$
• $22^2 = 484$

So, the hypotenuse is 22 cm.

Final Answer Calculation

Given Area $= 121 \text{ cm}^2$

Area $= \frac{1}{2} s^2 \implies s^2 = 2 \times 121 = 242$

Hypotenuse$^2 = s^2 + s^2 = 2s^2$

Hypotenuse$^2 = 2 \times 242 = 484$

Hypotenuse $= \sqrt{484} = 22$ cm

The hypotenuse of the isosceles right angled triangle is 22 cm.

Revision Table: Key Concepts for Triangle Calculations

Additional Information: Properties of Isosceles Right Triangles

An isosceles right triangle is also known as a 45-45-90 triangle because its angles are 45°, 45°, and 90°. There is a special relationship between the lengths of its sides:

• If the length of the equal sides (legs) is 's', then the length of the hypotenuse is $s\sqrt{2}$.

Let's check this with our result. We found $s = \sqrt{242} = 11\sqrt{2}$.

Hypotenuse $= s\sqrt{2} = (11\sqrt{2})\sqrt{2} = 11 \times (\sqrt{2})^2 = 11 \times 2 = 22$ cm.

This confirms our result obtained using the Pythagorean theorem directly from $s^2$. Understanding the 45-45-90 triangle properties can be a quicker way to solve such problems once the side 's' is found.