Solving for the Hypotenuse in Right Triangles: A Step-by-Step Guide

Understanding the properties of right triangles and how to find the length of the hypotenuse using the Pythagorean theorem is a fundamental concept in geometry. This article will walk you through various scenarios and provide step-by-step solutions to determine the hypotenuse when given different lengths for the legs of a right triangle.

Introduction to Right Triangles

A right triangle is a triangle with one angle equal to 90 degrees. In a right triangle, the Pythagorean theorem is used to find the length of the hypotenuse given the lengths of the other two sides. The theorem states that in a right triangle, the square of the hypotenuse (#961;) is equal to the sum of the squares of the other two sides (a and b):

Pythagorean Theorem: [ mathit{c}^2 mathit{a}^2 mathit{b}^2 ]

Solving for the Hypotenuse with Known Values

Example 1: Given Both Leg Lengths

For a right triangle where the shorter leg is 6 cm and the longer leg is 8 cm, we can use the Pythagorean theorem to find the hypotenuse:

Apply the Pythagorean theorem:

Pythagorean Theorem: [ mathit{c}^2 6^2 8^2 ]

Calculate the squares:

[ mathit{c}^2 36 64 ]

Simplify:

[ mathit{c}^2 100 ]

Taking the square root of both sides:

[ mathit{c} sqrt{100} ]

Thus, the hypotenuse is:

[ mathit{c} 10 text{ cm} ]

This is a simple illustration of how to find the hypotenuse when both legs' lengths are known.

Example 2: Given One Leg and One Leg's Relationship to Another

If the longer leg is 7 cm more than the shorter leg, let's denote the shorter leg as x cm. The longer leg will then be x 7 cm. The hypotenuse can be represented as:

Pythagorean Theorem: [ mathit{c}^2 x^2 (x 7)^2 ]

Expanding the equation:

[ mathit{c}^2 x^2 x^2 14x 49 ]

Simplify:

[ mathit{c}^2 2x^2 14x 49 ]

The exact value of c is:

[ mathit{c} sqrt{2x^2 14x 49} ]

Additional Scenarios and Solutions

Example 3: Given Specific Leg Lengths

Consider a triangle where one leg is 5 units and the other leg is 7 units. Using the Pythagorean theorem:

Pythagorean Theorem: [ mathit{c}^2 5^2 7^2 ]

Calculate the squares:

[ mathit{c}^2 25 49 ]

Simplify:

[ mathit{c}^2 74 ]

Take the square root of both sides:

[ mathit{c} sqrt{74} ]

Approximate this to:

[ mathit{c} approx 8.6 text{ units} ]

Example 4: Another Scenario with Known Relationships

Let's solve for a triangle where the smaller leg is L units, the longer leg is L 4 units, and the hypotenuse is L 8 units. Using the Pythagorean theorem:

Pythagorean Theorem: [ (L 8)^2 L^2 (L 4)^2 ]

Expand the equation:

[ L^2 16L 64 L^2 L^2 8L 16 ]

Simplify:

[ L^2 - 8L - 48 0 ]

Solve for L using the quadratic formula:

[ L frac{-(-8) pm sqrt{(-8)^2 - 4(1)(-48)}}{2(1)} ]

[ L frac{8 pm sqrt{64 192}}{2} ]

[ L frac{8 pm sqrt{256}}{2} ]

[ L frac{8 pm 16}{2} ]

Thus, L can be either 12 or -4. Since lengths cannot be negative, L 12 units.

The lengths are:

Shorter leg: 12 units

Longer leg: 16 units

Hypotenuse: 20 units

Conclusion and Summary

Understanding and applying the Pythagorean theorem to solve for the hypotenuse in right triangles is crucial for many mathematical and practical applications. By following the steps outlined in this article, you can easily solve for the hypotenuse in various scenarios where the lengths of the legs are known or related to each other.

Remember to always check your calculations to ensure accuracy and consistency. Whether you are dealing with straightforward values or more complex relationships, the Pythagorean theorem provides a reliable method for solving for the hypotenuse in right triangles.