Introduction

The Pythagorean Theorem is one of the most famous and interesting relationships in all of mathematics.  After years of observation and contemplation by ancient cultures, it was assumed and then proven that the area of the squares of the two shorter legs of a right triangle is equal to the area of the square of the hypotenuse of the triangle. 

The diagram to the right demonstrates the Pythagorean Theorem using areas.

The logical way to learn this concept is to first look at examples of right triangles where the lengths of the two legs is given.  Here is one such example:

Find the length of the hypotenuse of the right triangle to the left.

Solution:

The triangle is a right triangle, so the Pythagorean Theorem can be used to find the missing length.  a = 6 and b = 8, so a² + b² = 6² + 8² = 36 + 64 = 100.  c² = 100, and taking the square root of both sides yields the answer: c = 10.

Lesson

In the previous lesson on the Pythagorean Theorem, the lengths of the two legs were given and you were asked to find the hypotenuse of the triangle.  This process required you to square each length, add them up, and then take the square root of the total to find the exact length of the hypotenuse.

This lesson is very similar to the previous lesson in that you will still use the Pythagorean Theorem to find a missing length in a right triangle.  Instead of the unknown length being the hypotenuse, it is now one of the legs.  To find a missing leg, the first step is to square the length of the hypotenuse.  Second, square the length of the given leg and subtract it from the square of the hypotenuse.  Finally, take the square root of the difference in order to find the length of the missing leg.

Example 1:  Find the missing length in the right triangle using the Pythagorean Theorem.

Solution:

When finding the length of a missing leg, the process simply requires one additional subtraction as compared to finding the length of the hypotenuse.  The subtraction step is shown in blue in the solution above.

Example 2:  Find the missing length in the right triangle using the Pythagorean Theorem.

Solution:

Some teachers prefer to show the solution to examples 1 and 2 by altering the Pythagorean theorem and isolating the a² or the b².  The equation a² + b² = c² can be rewritten as a² = c² – b² or b² = c² – a².  Mathematically, this is the same thinking shown above, but with the equation altered to eliminate the subtraction step.  Example 3 shows how to do this.

Example 3:  Find the missing length of the triangle using (a variation of) the Pythagorean Theroem.

Solution:

Example 4:  Find the missing length of the triangle using (a variation of) the Pythagorean Theroem.

Solution:

Try It

Find the length of the missing leg in each right triangle:

 

The following right triangles are missing either a leg or a hypotenuse.  Find the length of the missing side in each triangle.

Solutions:

Find the length of the missing leg in each right triangle:

The following right triangles are missing either a leg or a hypotenuse.  Find the length of the missing side in each triangle.