# Lesson 14

Proving the Pythagorean Theorem

• Let’s prove the Pythagorean Theorem.

### 14.1: Notice and Wonder: Variable Version

What do you notice? What do you wonder?

### 14.2: Prove Pythagoras Right

Elena is playing with the equivalent ratios she wrote in the warm-up. She rewrites $$\frac{a}{x} = \frac{c}{a} \text{ as } a^2=xc$$. Diego notices and comments, “I got $$b^2=yc$$. The $$a^2$$ and $$b^2$$ remind me of the Pythagorean Theorem.” Elena says, “The Pythagorean Theorem says that $$a^2 + b^2 = c^2$$. I bet we could figure out how to show that.”

1. How did Elena get from $$\frac{a}{x} = \frac{c}{a} \text{ to } a^2=xc$$?
2. What equivalent ratios of side lengths did Diego use to get $$b^2=yc$$?
3. Prove $$a^2+b^2=c^2$$ in a right triangle with legs length $$a$$ and $$b$$ and hypotenuse length $$c$$.

### 14.3: An Alternate Approach

When Pythagoras proved his theorem he used the 2 images shown here. Can you figure out how he used these diagrams to prove $$a^2+b^2=c^2$$ in a right triangle with hypotenuse length $$c$$?

James Garfield, the 20th president, proved the Pythagorean Theorem in a different way.

• Cut out 2 congruent right triangles
• Label the long sides $$b$$, the short sides $$a$$ and the hypotenuses $$c$$.
• Align the triangles on a piece of paper, with one long side and one short side in a line. Draw the line connecting the other acute angles.

How does this diagram prove the Pythagorean Theorem?

### Summary

In any right triangle with legs $$a$$ and $$b$$ and hypotenuse $$c$$, we know that $$a^2+b^2=c^2$$. We call this the Pythagorean Theorem. But why does it work?

We can use an altitude drawn to the hypotenuse of a right triangle to prove the Pythagorean Theorem.

We can use the Angle-Angle Triangle Similarity Theorem to show that all 3 triangles are similar. Because the triangles are similar, corresponding side lengths are in the same proportion.

Because the largest triangle is similar to the smaller triangle, $$\frac{c}{a}=\frac{a}{d}$$. Because the largest triangle is similar to the middle triangle, $$\frac{c}{b}=\frac{b}{e}$$. We can rewrite these equations as $$a^2=cd$$ and $$b^2=ce$$.

We can add the 2 equations to get that $$a^2+b^2=cd+ce$$ or $$a^2+b^2=c(d+e)$$. From the original diagram we can see that $$d+e=c$$, so $$a^2+b^2=c(c)$$ or $$a^2+b^2=c^2$$.

Using the Pythagorean Theorem we can describe a triangle's angles without ever drawing it. For example, a triangle with side lengths 8, 15, and 17 is right because $$17^2=8^2+15^2$$. A triangle with side lengths 8, 15, and 18 is obtuse because $$18^2>8^2+15^2$$. A triangle with side lengths 8, 15, and 16 is acute because $$16^2<8^2+15^2$$.

### Glossary Entries

• altitude

An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.