Finding the Missing Leg of a Right Triangle Using the Pythagorean Theorem

Introduction to the Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of geometry, providing a method to find the length of a missing side in a right triangle. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is expressed as:

c^2  a^2   b^2

Using the Pythagorean Theorem to Find the Missing Leg

Given a right triangle where the hypotenuse measures 10 meters and one leg measures 8 meters, we can utilize the Pythagorean Theorem to determine the length of the other leg. Let's denote the hypotenuse as 'c', and the two legs as 'a' and 'b'. Given:

hypotenuse c 10 meters

one leg a 8 meters

unknown leg b needs to be determined

The mathematical setup based on the Pythagorean Theorem is:

10^2  8^2   b^2

Let's break down the steps:

Input the known measurements:

10^2 8^2 b^2

100 64 b^2

Isolate b^2 by subtracting 64 from both sides:

100 - 64 b^2

36 b^2

Solve for b by taking the square root of both sides:

b √36 6 meters

Thus, the length of the other leg is 6 meters.

Another Perspective: Identifying a Pythagorean Triple

In another solution, the right triangle is identified as a 3-4-5 triangle scaled by a factor of 2. This means the sides of the triangle are in the ratio 3:4:5. Given:

hypotenuse 10 meters (5*2)

one leg 8 meters (4*2)

the other leg 6 meters (3*2)

This indicates the missing leg is 6 meters.

Final Solution: Using Algebra

A different approach involves setting up the equation and simplifying:

Set up the equation based on the Pythagorean Theorem:

10^2 6^2 x^2

100 36 x^2

x^2 64

x √64 8 meters

This reaffirms that the length of the other leg is 8 meters.

Conclusion

The Pythagorean Theorem provides a straightforward method to find the missing side of a right triangle. Whether by direct calculation, identifying a Pythagorean triple, or algebraic simplification, the consistent application of the theorem leads to the same result. In the examples provided, the missing leg is either 6 meters or 8 meters, depending on the specific configuration and the steps taken in the calculation. Understanding and applying the Pythagorean Theorem is fundamental in trigonometry, architecture, and a myriad of other practical applications.