Finding the Length of the Other Leg of a Right Triangle Using the Pythagorean Theorem

Understanding the relationship between the sides of a right triangle is fundamental in various fields, including mathematics, engineering, and architecture. One of the most renowned principles used for solving these problems is the Pythagorean Theorem. This article will guide you through the process of determining the length of the other leg of a right triangle when the length of one leg and the hypotenuse are provided.

The Pythagorean Theorem in Action

Let's start with a classic example. Imagine a right triangle with a hypotenuse of 5 inches and one leg of 3 inches. How would we find the length of the other leg? The Pythagorean Theorem states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): c2 a2 b2. In this case, we are given the hypotenuse (c) and one leg (a), and we need to find the other leg (b).

Step-by-Step Solution

First, we write down the given values: c 5 inches (hypotenuse) and a 3 inches (one leg).

Next, we use the Pythagorean Theorem, substituting the known values: 52 32 b2.

Calculate the squares: 25 9 b2.

Isolate b2 by subtracting 9 from both sides: 25 - 9 b2.

Perform the subtraction: 16 b2.

Finally, take the square root of both sides to find b: b √16 4 inches.

Thus, the length of the other leg is 4 inches. You may have noticed that this forms a 3-4-5 right triangle, a commonly used and easily recognizable pythagorean triplet.

General Approach to Finding the Other Leg

When dealing with any right triangle where the hypotenuse (h) and one leg (L1) are known, you can use the Pythagorean Theorem to find the other leg (L2). The formula is:

h2 L12 L22

By rearranging this equation, you get:

L2 √(h2 - L12)

This formula allows you to calculate the length of the unknown leg directly.

Interactive Example

Let's solve a more general problem: If the hypotenuse is 8 inches and the known leg is 5 inches, what is the length of the other leg?

Write down the equation: 82 52 L22.

Calculate the squares: 64 25 L22.

Isolate L22 by subtracting 25 from both sides: 64 - 25 L22.

Perform the subtraction: 39 L22.

Take the square root of both sides: L2 √39 approx; 6.24 inches.

The length of the other leg is approximately 6.24 inches.

Conclusion

The Pythagorean Theorem is a powerful tool for solving right triangles. By understanding and applying this theorem, you can easily find the length of the unknown leg when you know the hypotenuse and one leg. This knowledge has practical applications in many fields and is a fundamental concept in geometry and trigonometry.

If you need more practice or have any specific questions, feel free to ask, and I'll be happy to help!