The Pythagorean Theorem Study Guide (page 2)
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.
If a right triangle has a hypotenuse length of 9 feet and a leg length of 5 feet, what is the length of the third side? We use the Pythagorean theorem with the hypotenuse, 9, by itself on one side, and the other two lengths, 5 and x, on the other.
52 + x2 = 92
x = √81 – 25 = √56 = 2√14 ≈ 7.48
The third side is about 7.48 feet long.
Pythagorean Word Problems
Many word problems involve finding a length of a right triangle. Identify whether each given length is a leg of the triangle or the hypotenuse. Then solve for the third length with the Pythagorean theorem.
A diagonal board is needed to brace a rectangular wall. The wall is 8 feet tall and 10 feet wide. How long is the diagonal?
Having a rectangle means that we have a right triangle, and that the Pythagorean theorem can be applied. Because we are looking for the diagonal, the 10-foot and 8-foot lengths must be the legs, as shown in Figure 3.19.
The diagonal D must satisfy the Pythagorean theorem:
D2 = 102 + 82
D2 = 100 + 64 = 164
D = √164 = 2√41 ≈ 12.81 feet
A 100-foot rope is attached to the top of a 60-foot tall pole. How far from the base of the pole will the rope reach?
We assume that the pole makes a right angle with the ground, and thus, we have the right triangle depicted in Figure 3.20. Here, the hypotenuse is 100 feet.
The sides must satisfy the Pythagorean theorem:
x2 + 602 = 1002
x = √6,400 = 80 feet
Since the Pythagorean theorem was discovered, people have been especially fascinated by right triangles with whole number sides. The most famous one is the 3-4-5 right triangle, but there are many others, such as 5-12-13 and 6-8-10. A Pythagorean triple is a set of three whole numbers a-b-c with a2 + b2 = c2. Usually, the numbers are put in increasing order.
Is 48-55-73 a Pythagorean triple?
We calculate 482 + 552 = 2,304 + 3,025 = 5,329. It just happens that 732 = 5,329. Thus, the numbers 48, 55, and 73 form a Pythagorean triple.
Generating Pythagorean Triples
The ancient Greeks found a system for generating Pythagorean triples. First, take any two whole numbers r and s, where r > s. We will get a Pythagorean triple a-b-c if we set:
a = 2rs
b = r2 – s2
c = r2 + s2
This is because
a2 + b2
= (2rs)2 + (r2 – s2)2
= 4r2s2 + r4 – 2r2s2 + s4
= r4 + 2r2s2 + s4
= (r2 + s2)2 = c2
Thus, a2 + b2 = c2.
Find the Pythagorean triple generated by r = 7 and s = 2.
a = 2rs = 2(7)(2) = 28
b = r2 – s2 = 72 – 22 = 45
c = r2 + s2 = 72 + 22 = 53
Thus, r = 7 and s = 2 generate the 28-45-53 Pythagorean triple.
If we plug in different values of r and s, we will generate many different Pythagorean triples, whole numbers a, b, and c with a2 + b2 = c2. Around 1640, a French mathematician named Pierre de Fermat suggested that this could happen only with the exponent 2. He said that no positive whole numbers a, b, and c could possibly make a3 + b3 = c3 or a4 + b4 = c4 or an + bn = cn for any n > 2. Fermat claimed to have found a short and clever proof of this, but then died without writing it down.
For hundreds of years, mathematicians tried to prove this result, called "Fermat's Last Theorem." Only in 1995 was it finally proven, by a British mathematician named Andrew Wiles. Because his proof runs to more than 100 pages, it is clearly not the simple proof that Fermat spoke about. This assumes, of course, that Fermat had actually found a correct and simple proof.
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