Introduction

When working with triangles, particularly isosceles right triangles, knowing how to calculate the hypotenuse can be incredibly useful for solving various geometric problems. In this article, we will walk through the process of finding the hypotenuse of an isosceles right triangle given its area. We'll explore the Area Formula, the Pythagorean Theorem, and then apply these concepts to solve a practical example.

Understanding the Area of an Isosceles Right Triangle

An isosceles right triangle is a special type of right triangle in which two sides (legs) are congruent. The area of such a triangle can be calculated using a specific formula:

Area Formula

The area (A) of an isosceles right triangle can be expressed as:

A frac{1}{2} times b times h

However, in an isosceles right triangle, the base and height are equal, and we can denote both as (a). Therefore, the area can be simplified to:

A frac{1}{2} times a times a frac{1}{2} a^2

This formula allows us to connect the area of the triangle to the length of the legs.

Given Area: 121 cm2

In this specific example, we are given that the area of the isosceles right triangle is 121 cm2. We can set up the equation and solve for (a), the length of the legs, as follows:

frac{1}{2} a^2 121

Solving for (a^2)

First, we need to solve for (a^2):

a^2 121 times 2 242

Finding (a)

Next, we find the value of (a):

a sqrt{242} sqrt{121 times 2} 11 sqrt{2} text{ cm}

Here, we used the property of square roots to break down (sqrt{242}) into (sqrt{121 times 2}).

Calculating the Hypotenuse

The hypotenuse (c) of an isosceles right triangle can be found using the Pythagorean theorem:

c sqrt{a^2 a^2} sqrt{2a^2} a sqrt{2}

Substituting the value of (a):

c 11 sqrt{2} times sqrt{2} 11 times 2 22 text{ cm}

Therefore, the length of the hypotenuse is 22 cm.

Conclusion

By understanding the area formula and the Pythagorean theorem, we can easily find the hypotenuse of an isosceles right triangle given its area. The key steps involve setting up the area equation, solving for the leg length, and then using the Pythagorean theorem to find the hypotenuse.