Lesson 14

Proving the Pythagorean Theorem

  • Let’s prove the Pythagorean Theorem.

14.1: Notice and Wonder: Variable Version

Triangle with sides a, b, and c. Right angle between a and b. Segment h drawn from vertex between a and b and meets c at a right angle. H splits c into 2 lengths labeled x and y. X is adjacent to a.

What do you notice? What do you wonder?

14.2: Prove Pythagoras Right

Triangle with sides a, b, and c. Right angle between a and b. Segment h drawn from vertex between a and b and meets c at a right angle. H splits c into 2 lengths labeled x and y. X is adjacent to a.

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

Diagram of 2 congruent squares.

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?

Diagram with 2 paper triangles.

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.

Right triangle a b c.

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.

3 similar triangles. The largest triangle has sides a, c, b. The middle-sized has sides f, b, e. The smallest has sides d, a, f.

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.