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# Geometry: The Pythagorean Theorem

## The Pythagorean Theorem

Now that you know how to show that two triangles are similar, you can use CSSTAP to find relationships between the sides of similar triangles. You can even create the theorems necessary to prove one of the most famous theorems in geometry: the Pythagorean Theorem. But before you can tackle the Pythagorean Theorem, you'll need a theorem about altitudes. I'll walk through an explanation of why the theorem is true, but I will not write out a formal proof.

• Theorem 14.1: The altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original right triangle.

Figure 14.1 will help clarify what is going on. The altitude drawn to the hypotenuse has to originate at the vertex of the right angle (C) and is perpendicular to the hypotenuse (AB). Let's call the point where the altitude and the hypotenuse intersect D. You have three triangles to relate: the original triangle ?ABC and two new, smaller triangles ?ACD and ?CBD. Remember, you have to be careful about the order of the vertices. You have to match up corresponding vertices when representing the similarity of two triangles. You can use the AA Similarity Theorem and the transitive property of similarity (remember, similarity is an equivalence relation) to show that all three triangles are similar. Figure 14.1?ABC with altitude CD from C to the hypotenuse AB.

First, I'll show that ?ABC and ?CBD are similar. Both triangles are right triangles, so there's one pair of congruent angles. As for the second angle, notice that ?B is involved in both triangles. Because ?B ~= ?B, you've got a second pair of congruent angles, and by the AA Similarity Theorem, ?CBD ~= ?ABC. You need to match up your angles and your vertices: ?D is the right angle in ?CBD, which corresponds to ?C in ?ABC, ?B corresponds to itself, and ?BCD corresponds to ?A.

You'll use a similar argument to show that ?ABC and ?ACD are similar. ?D is the right angle in ?ACD, so it corresponds to ?C in ?ABC. ?A corresponds to itself, which leaves ?ACD and ?B to correspond.

Because ?ACD ~ ?ABC and ?ABC ~ ? CBD , the transitive property of ~ shows that ?ACD ~ ?CBD. Because CSSTAP, you know that

This should look a bit familiar. From this proportionality, you see that CD is the geometric mean of the lengths of the segments of the hypotenuse! The Pythagorean Theorem is just a couple of algebraic steps away.

Now that you know these three triangles are similar, you can break them apart to get a better handle on the proportionalities involved. You might want to refer to Figure 14.2. Figure 14.2?ABC can be split into two triangles: ?ACD and ?CBD. All three triangles are similar.

Because ?ACD ~ ?ABC, you know that

Because ?ABC ~ ?CBD , you know that

• AB/BC = BC/BD.

If you cross-multiply, add the two equations together, and simplify, you will derive the Pythagorean Theorem! I'll write out the details in a two-column proof.

• The Pythagorean Theorem: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs.

Translating the Pythagorean Theorem into the drawing in Figure 14.2, the goal is to prove that (AB)2 = (BC)2 + (AC)2.

Statements Reasons
1. ?ACD ~ ?ABC and ?ABC ~ ?CBD Previous discussion
2. AB/AC = AC/AD and AB/BC = BC/BD CSSTAP