Pythagorean Theorem
In this lesson, one of the most famous theorems in all of mathematics will be discussed. The Pythagorean Theorem demonstrates a relationship among the three sides of a right triangle. Proof of this theorem will be given.
The Pythagorean Theorem states that the sum of the areas of the squares on the two legs of a right triangle is equal to the area of a square on the hypotenuse (the side opposite the 90º or right angle). The theorem is stated two different ways and illustrated below:

1. Using Leg 1 and Leg 2 to mean the lengths of the sides forming the right angle, and Hyp to mean the length of the hypotenuse, we have
2. Squares have been drawn on each side of the triangle. These squares are numbered I, II, and III. The areas of these squares are related as follows: Area I + Area II = Area III
The converse of the Pythagorean Theorem is also true: If three sides of length x, y, and q are the lengths of the sides of a triangle and it is true that , then the triangle is a right triangle and q is the hypotenuse.
Uses of this theorem A typical application of this theorem is to find the measure of one side of a right triangle when the measure of the other two sides are known.
1. Suppose that the measures of the two legs of a right triangle are 3 meters and 4 meters. What is the measure of the hypotenuse? A diagram and solution are shown below.

We set up an equation from the diagram at right using the fact that

We have . This is the same as or . Therefore, x = 5.

The lengths of these sides are 3, 4, and 5. This is known as a Pythagorean triple: all the sides have lengths which are whole numbers. Two other Pythagorean triples are: 5, 12, and 13 as well as 8, 15, and 17. Note that multiples of these integers form Pythagorean triples and therefore lengths of sides of right triangles. For example, 6, 8, and 10 as well as 16, 30, and 34 are both Pythagorean triples. There are infinitely many Pythagorean triples, such that the sides of a right triangle are whole numbers. Another example is shown below.
1. Suppose that the measure of one leg of a right triangle is 24 meters and the hypotenuse is 30 meters. What is the measure of the second leg?

Using the definition for the triangle shown above, we have . Simplifying this equation results in . Solving for x we get the value  meters.
Of course, it is not always true that all three sides are of whole number lengths. In the examples that follow be ready to work with square roots of numbers as you solve these equations. For help in working with square roots, reference the lesson on simplifying radicals. In the following examples, we will not use units such as meters or inches, we will just use numbers to represent the lengths of the sides.

Let's practice:
1. What is the value of x in the triangle shown below?

and so for the diagram below we have This gives
1. What is the value of a in the triangle shown below?

and so from the next diagram given below we have . Therefore
1. What is the value of a in the triangle shown below?

and so from the diagram given below we have . This is the same as . We have .
1. Is the triangle given below a right triangle?
 diagonal = 7 5 2
Looking at the measurements, we try squaring the lengths of the two smaller sides and adding them to see if they equal the square of the longest side. We have . Therefore, this triangle is NOT a right triangle.

Examples
 If the three sides of a triangle are of length 27, 36, and 45 is the triangle a right triangle? What is your answer?
 If the three sides of a triangle are of length 7, 9, and 12 is the triangle a right triangle? What is your answer?
 If two legs of a right triangle are of length 3 and 7, what is the length of the hypotenuse? What is your answer?
 If one leg of a right triangle is 7 and the hypotenuse is 21, what is the length of the other leg of this triangle? What is your answer?
 If the two legs of a right triangle are , what is the length of the hypotenuse? What is your answer?

M Ransom

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