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## Introduction

Mathematicians have worked for centuries to uncover relationships in triangles and other polygons.  One of the most famous and helpful relationships was discovered by a Greek mathematician named Pythagorus.  He discovered that the three sides of a right triangle were related in the following way:

If the lengths of each of the shorter two sides (legs) of a right triangle are squared and added together, the total is equal to the length of the third side (hypotenuse) squared.

So any time you see a right triangle, remember that the lengths of the two shorter sides are related to the length of the longest side.

## Lesson

If you had the time to turn each side of a right triangle into a square, you would find that the two smaller squares have an interesting relationship in comparison to the larger square.

The 3-4-5 triangle shown to the right is one of the most famous triangles in all of mathematics.  Squaring each of the two shorter sides yields 32 + 42 = 9 + 16 = 25.  The longer side is 5 units,   and 52 = 25.  There are a total of 25 red squares and 25 blue squares in the picture.  This relationship holds true for any right triangle.

In a right triangle, the two shorter sides are called legs and the longest side is called the hypotenuse.  Usually a is the shorter of the two legs and b is the longer of the two legs.  In some cases, a is the same length is b.  All right triangles contain a longest side that is directly across from the right angle.  This longest side is represented by the variable “c” and is directly across from the right angle.

Example 1:  Use the Pythagorean Theroem to find the length of the missing side in the triangle.

Solution:

Example 2:  Find the missing length of the triangle using the Pythagorean Theroem.

Solution:

Many teachers begin teaching this concept using triangles whose sides are all whole numbers.  Since the final step in finding the missing side is taking a square root, most answers do not work out evenly.  When a problem contains a triangle whose side lengths are all whole numbers, the answer is often one of the following:

Other right triangle comparisons are simply variations of the above common right triangles.  For example, the triangle whose sides measure 6-8-10 is a right triangle because its sides have the same ratio as the “common” right triangle 3-4-5.

One easy way to tell that you are dealing with a variation of a “common” right triangle is to compare the lengths of the three sides and reduce their ratios to lowest terms.  For example, if the lengths of a triangle measure 14, 48, and 50, each of these numbers is divisible by 2.  The triangle 14-48-50 is a variation of a 7-24-25 triangle.

Remembering the lengths of common right triangles and being able to analyze their variations can speed up the process of determining whether three lengths form a right triangle.  In example 3, analyze the three given lengths and determine if they form a common right triangle.

Example 3:  Determine whether the three lengths form a right triangle.  If possible, reduce the lengths to one of the common right triangles given above.

• Lengths of 15, 20, and 25.
• Lengths of 40, 96, and 104
• Lengths of 60, 63, and 80
• Lengths of 36, 39, and 15
• Lengths of 14, 24, and 26
• Lengths of 34, 16, and 30

Solution:

Lengths of 15, 20, and 25

• Each length is divisible by 5, so this triangle’s lengths can be reduced to 3, 4, and 5.  This is a common right triangle, so our original triangle is a right triangle.

Lengths of 40, 96, and 104

• Each length is divisible by 8, so this triangle’s lengths can be reduced to 5, 12, and 13.  This is a common right triangle, so our original triangle is a right triangle.

Lengths of 60, 63, and 80

• There is no common factor for these three numbers, so the Pythagorean theorem must be used to evaluate the lengths.  The shorter two sides are 60 and 63, and 602 + 632 = 3600 + 3969 = 7569.  The longest side is 80, and 802 = 6400.  Since the totals are different (7569 ≠ 6400), the triangle is not a right triangle.

Lengths of 36, 39, and 15

• Each length is divisible by 3, so this triangle’s lengths can be reduced to 12, 13, and 5.  The lengths aren’t listed in ascending order, and reordering the lengths yields 5, 12, and 13.  This is a common right triangle, so our original triangle is a right triangle.

Lengths of 14, 24, and 26

• Each lengths is divisible by 2, so this triangle’s lengths can be reduced to 7, 12, and 13.  This is not a common right triangle, so the Pythagorean theorem must be used to evaluate the lengths.  The shorter two sides are 14 and 24, and 142 + 242 = 196 + 576 = 772.  The longest side is 26, and 262 = 676.  Since the totals are different (772 ≠ 676), the triangle is not a right triangle.

Lengths of 34, 16, and 30

• Each length is divisible by 2, so this triangle’s lengths can be reduced to 17, 8, and 15.  This is a common right triangle, so our original triangle is a right triangle.

Most students prefer dealing with whole numbers and avoiding decimals whenever possible.  The common right triangles above are examples of triangles whose sides are all whole numbers.  A more comprehensive list of useful right triangles can be found here:

List of Pythagorean Triples

Although teacher and textbooks use many examples that work out even, chances are that the hypotenuse may work out to be a decimal even if the two legs have whole number lengths.  Example #4 and #5 show how this is possible.

Example 4:  Use the Pythagorean Theroem to find the length of the missing side in the triangle.

Solution:

Example 5:  Find the missing length of the triangle using the Pythagorean Theroem.

Solution:

## Try It

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

Determine whether each triangle is one of the “common” right triangles presented in this lesson.  If not, then use the Pythagorean Theorem to tell whether the triangle is a right triangle or not.

5)  Side lengths: 30, 40, and 50

6)  Side lengths: 18, 27, 36

7)  Side lengths: 24, 45, 51

8)  Side lengths: 10, 10, 20

9)  Side lengths: 14, 48, 50

10) Side lengths:  30, 80, 89

Solutions:

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

Determine whether each triangle is one of the “common” right triangles presented in this lesson.  If not, then use the Pythagorean Theorem to tell whether the triangle is a right triangle or not.

5)  Side lengths: 30, 40, and 50

Each length is divisible by 10, so this triangle’s lengths can be reduced to 3, 4, and 5.  This is a common right triangle, so our original triangle is a right triangle.

6)  Side lengths: 18, 27, 36

Each lengths is divisible by 9, so this triangle’s lengths can be reduced to 2, 3, and 4.  This is not a common right triangle, so the Pythagorean theorem must be used to evaluate the lengths.  The shorter two sides are 2 and 3, and 22 + 32 = 4 + 9 = 13.  The longest side is 4, and 42 = 16.  Since the totals are different (13 ≠ 16), the triangle is not a right triangle.

7)  Side lengths: 24, 45, 51

Each length is divisible by 3, so this triangle’s lengths can be reduced to 8, 15, and 17.  This is a common right triangle, so our original triangle is a right triangle.

8)  Side lengths: 10, 10, 20

Each lengths is divisible by 10, so this triangle’s lengths can be reduced to 1, 1, and 2.  This is not a common right triangle, so the Pythagorean theorem must be used to evaluate the lengths.  The shorter two sides are 1 and 1, and 12 + 12 = 1 + 1 = 2.  The longest side is 2, and 22 = 4.  Since the totals are different (2 ≠ 4), the triangle is not a right triangle.

9)  Side lengths: 14, 48, 50

Each length is divisible by 2, so this triangle’s lengths can be reduced to 7, 24, and 25.  This is a common right triangle, so our original triangle is a right triangle.

10) Side lengths:  39, 80, 89

The lengths do not have a common factor and therefore cannot be reduced.  These lengths do not form a common right triangle, so the Pythagorean theorem must be used to evaluate the lengths.  The shorter two sides are 39 and 80, and 392 + 802 = 1521 + 6400 = 7921.  The longest side is 89, and 892 = 7921.  The totals are the same (7921 = 7921), so the triangle is a right triangle.