The Pythagorean Theorem Study Guide
The Pythagorean Theorem
In this lesson, we examine a very powerful relationship between the three lengths of a right triangle. With this relationship, we will find the exact length of any side of a right triangle, provided we know the lengths of the other two sides. We also study right triangles where all three sides have whole number lengths.
Proving the Pythagorean Theorem
The Pythagorean theorem is this relationship between the three sides of a right triangle. Actually, it relates the squares of the lengths of the sides. The square of any number (the number times itself) is also the area of the square with that length as its side. For example, the square in Figure 3.1 with sides of length a will have area a · a = a2.
The longest side of a right triangle, the side opposite the right angle, is called the hypotenuse, and the other two sides are called legs. Suppose a right triangle has legs of length a and b, and a hypotenuse of length c, as illustrated in Figure 3.2.
The Pythagorean theorem states that a2 + b2 = c2, This means that the area of the squares on the two smaller sides add up to the area of the biggest square. This is illustrated in Figures 3.3 and 3.4.
This is a surprising result. Why should these areas add up like this? Why couldn't the areas of the two smaller squares add up to a bit more or less than the big square?
We can convince ourselves that this is true by adding four copies of the original triangle to each side of the equation. The four triangles can make two rectangles, as shown in Figure 3.5. They could also make a big square with a hole in the middle, as in Figure 3.6.
If we add the a2 and b2 squares to Figure 3.4, and the c2 square to Figure 3.5, they fit exactly. The result in either case is a big square with each side of length a + b, as shown in Figure 3.7.
The two big squares have the same area. If we take away the four triangles from each side, we can see that the two smaller squares have the exact same area as the big square, as shown in Figure 3.8. Thus, a2 + b2 = c2
This proof of the Pythagorean theorem has been adapted from a proof developed by the Chinese about 3,000 years ago. With the Pythagorean theorem, we can use any two sides of a right triangle to find the length of the third side.
Suppose the two legs of a right triangle measure 8 inches and 12 inches, as shown in Figure 3.9. What is the length of the hypotenuse?
By the Pythagorean theorem:
122 + 82 = H2
208 = H2
H = √208
While the equation gives two solutions, a length must be positive, so H = √208. This can be simplified to H = 4√13.
What is the height of the triangle in Figure 3.10?
Even though the height is labeled h, it is not the hypotenuse. The longest side has length 10 feet, and thus must be alone on one side of the equation.
h2 + 32 = 102
h2 = 100 – 9
h = √91 ≈ 9.54
With the help of a calculator, we can see that the height of this triangle is about 9.54 feet.
We can use the Pythagorean theorem on triangles without illustrations. All we need to know is that the triangle is right and which side is the hypotenuse.
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