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## An Introduction to the Pythagorean Theorem

Mathematicians have labored for ages to discover the relationships within triangles to other polygons. A famous and useful relationship was discovered by a Greek mathematician called Pythagoras. He found that the sides of a right triangle are related in the following way:

When the lengths of each of the smaller two sides (legs) of a right triangle are squared and the squares are added together, the total length of the 3rd side (called the hypotenuse) is equal to the square. So when you see a right triangle, keep in mind that the length of the 2 shorter sides is related to the length of the longest side.

If a person has time to make three exterior squares from each side of any right triangle, you will find that smaller squares result in an interesting relationship compared to larger squares.

A triangle whose sides measure 3 units, 4 units, and then 5 units is one of the most famous triangles in all of mathematics. Squaring each of the 2 smaller sides yields 25 square units equal to nine sixteens. The long side is five units, which means its square area is twenty-five square units. This arrangement is true for each and every right triangle.

Many problems that deal with right triangles give decimal answers. However, there are many examples of combinations of whole numbers possible in right triangles. Some of these are:

Three, four, five

Six, eight, ten

Five, twelve, thirteen

Seven, twenty-four, twenty-five

Eight, fifteen, seventeen

Each combination above represents three lengths of a right triangle. In the classroom, teachers often begin teaching the concept of the Pythagorean theorem using whole number examples. Later in the class, many answers may have one or more sides whose lengths are not whole numbers.

In a right triangle, the 2 shorter sides are the legs and the longest side is known as the hypotenuse. Usually a stands as the shorter of two feet and b is usually a foot longer. In some cases, a is the same length as b. All right triangles have the longest side that crosses directly through the right angle. This longest side is represented by the variable “c” and is referred to as the hypotenuse.

Often, major discoveries in science and mathematics receive specific names. Since this particular relationship within a right triangle was discovered by Pythagoras, it is called the Pythagorean Theorem in his honor.

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