Lesson 14

What do you notice? What do you wonder?

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 \(c\).

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 \(c\)?

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 \(16^2<8^2+15^2\).