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## An Introduction to the Pythagorean Theorem

Mathematicians have struggled throughout the ages to discover relationships within triangles and other polygons. One famous and useful relationship was discovered by the Greek mathematician Pythagorus. He found that the sides of a right triangle are related as follows:

If the lengths of both shorter sides (legs) of a right triangle are squared and the squares added together, the total is the same as the length of the 3rd side (called the hypotenuse) squared. So, when you notice a right triangle, remember that the lengths of the two shorter sides are related to the length of the longer side.

If a person had time to form the three outer squares from each side of any right triangle, you will find that the smaller squares give an interesting relationship compared to the large square.

A triangle with sides measuring 3 units, 4 units, and then five units is one of the most familiar triangles in almost all of mathematics. Squaring both smaller sides gives nine sixteen equals 25 square meters. The longer side is five units, which means that the area of this square is twenty-five square units. This arrangement applies to any right triangle.

Many problems involving right triangles give decimal answers. However, there are many examples of integer combinations possible in right triangles. Some of them are:

three, four, five

six, eight, ten

five, twelve, thirteen

seven, twenty four, twenty five

eight, fifteen, seventeen

All of the above combinations represent the three lengths of a right triangle. In class, teachers often start teaching the concept of the Pythagorean theorem using integer examples. Later in the class, many answers may contain one or more sides whose lengths are not integers.

A right triangle has 2 smaller sides as legs and the longest side is called the hypotenuse. Usually a stands out as the shorter of the two legs and b is usually the longer leg. In some cases, a is identical in length to b. All right triangles contain a longest side that is directly opposite the right angle. This longest side is represented by the variable “c” and is called the hypotenuse.

Often, major discoveries in science and mathematics are given distinctive names. Because Pythagoras discovered this special relationship within right triangles, it has been named the Pythagorean Theorem in his honor.

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