Math 106 - Cooley Math For Elementary Teachers II OCC

Activity #16 – The Pythagorean Theorem

California State Content Standard – Measurement and Geometry for Grade Seven
1.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by
constructing figures that meet given conditions and by identifying attributes of figures:
3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a
right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem
by direct measurement.

The Pythagorean Theorem is one of the earliest know theorems to ancient civilizations. It was named after Pythagoras, a Greek mathematician and philosopher. The theorem bears his name although we have evidence that the Babylonians knew this relationship some 1000 years earlier. Plimpton 322, a Babylonian mathematical tablet dated back to 1900 B.C., contains a table of Pythagorean triples. The Chou-pei, an ancient Chinese text, also gives us evidence that the Chinese knew about the Pythagorean theorem many years before Pythagoras or one of his colleagues in the Pythagorean society discovered and proved it. This is the reason why the theorem is named after Pythagoras.

The Pythagorean Theorem

In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then,

.

Converse of the Pythagorean Theorem

Let a triangle have sides of length a, b, and c. If , then the triangle is a right triangle and the angle opposite the side of length c is its right angle..

Pythagorean Triples

A Pythagorean triple is a set of three integers a, b, and c which form the sides of a right triangle.

There are infinitely many Pythagorean triples. There are 50 with a hypotenuse less than 100 alone. Here are the first few: 3:4:5 , 5:12:13 , 7:24:25 , 8:15:17 , 9:40:41 , 11:60:61, etc.

If you take any Pythagorean triple and multiply each number by some same integer, the result will be another Pythagorean triple. This accounts for why there is an infinite number of them. So, if you take the first Pythagorean triple 3:4:5, and multiply by an integer, let’s say 2, we have a new Pythagorean triple of 6:8:10. Memorizing some common Pythagorean triples can be very helpful.


Proof of the Pythagorean Theorem (Algebraic Method):

Start out with the following figure:

The illustration shows a large square with identical right triangles in each of its four corners along with a smaller square having side of length c.

The area of each right triangle is .

The area of the smaller square is .

Thus, the area of the larger square is .

However, as the large square has sides of length a + b, we can calculate its area which is .

So, setting the two areas equal to each other, we have

Proof of the Pythagorean Theorem (Geometric Method):

Here is another way to look at:

Notice the area of the inner square in yellow is c2.

Label the right triangles as shown in Figure 1.

Figure 1 Figure 2 Figure 3 Figure 4

First, move Triangle 1 along the inner square, as shown in Figure 2.

Then, move Triangle 2 to the left, as shown in Figure 3.

Now, move Triangle 3 straight up, as shown in Figure 4.

Thus, we have the following illustration:

Here, you can clearly see, that the yellow area that was once c2, is now in two different pieces. These pieces have area a2 and b2. Thus,


J Exercises: Find the length of the third side of each triangle. If an answer is not a whole number, use

radical notation to give an exact answer and decimal notation for an approximation to the

nearest thousandth.

1) 2)

3) 4)

5) 6)

7) If a and b are the legs and c is the hypotenuse of a right triangle, what is the length of b, if a = 3

and c = ?

- 1 -