Finding the Shortest Leg of a Right Triangle Using the Pythagorean Theorem: A Step-by-Step Guide

In this article, we will explore how to find the length of the shortest leg of a right triangle when given specific information about the other leg and the hypotenuse. We will use the Pythagorean theorem to solve a quadratic equation, providing a step-by-step solution for better understanding.

Understanding the Problem

Consider a right triangle where one leg is 28 millimeters shorter than the other leg, and the hypotenuse is 68 millimeters. Let's denote the length of the shortest leg as x. Consequently, the other leg will be x 28. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, this can be expressed as:

[ left( x right)^2 left( x 28 right)^2 68^2 ]

Expanding and simplifying this equation, we get:

[ x^2 x^2 56x 784 4624 ] [ 2x^2 56x 784 - 4624 0 ] [ 2x^2 56x - 3840 0 ] [ x^2 28x - 1920 0 ]

Solving the Quadratic Equation

Now, we need to solve the quadratic equation [ x^2 28x - 1920 0 ]. There are several methods to solve this, including factoring, completing the square, or using the quadratic formula. In this case, we will use the quadratic formula:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Substituting the values [ a 1, b 28, c -1920 ], we get:

[ x frac{-28 pm sqrt{28^2 - 4 cdot 1 cdot (-1920)}}{2 cdot 1} ] [ x frac{-28 pm sqrt{784 7680}}{2} ] [ x frac{-28 pm sqrt{8464}}{2} ] [ x frac{-28 pm 92}{2} ]

This gives us two solutions:

[ x frac{64}{2} 32 ] [ x frac{-120}{2} -60 ]

Since a leg length cannot be negative, we discard the negative solution. Therefore, [ x 32 ].

Conclusion

The length of the shortest leg in this right triangle is 32 millimeters. The other leg, which is 28 millimeters longer, is:

[ x 28 32 28 60 ]

We can verify our solution using the Pythagorean theorem:

[ 32^2 60^2 1024 3600 4624 68^2 ]

Therefore, the length of the shortest leg is 32 millimeters.

Additional Information

Understanding the Pythagorean theorem is essential in solving problems involving right triangles. It is commonly used in fields such as architecture, engineering, and construction. Practicing similar problems can help you become more proficient in applying the theorem.

Practice Problems

Try solving the following problem using the same method:

A right triangle has a hypotenuse of 68 millimeters, and the other leg is 32 millimeters shorter than the hypotenuse. Find the lengths of the two legs.

By following the steps outlined in this article, you can confidently solve such problems and gain a better understanding of the Pythagorean theorem.