The Pythagorean Theorem

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In this activity we will prove the most famous theorem of all.

Remind us, what is the most famous theorem of all and what exactly does it assert?

Euclid’s proof

What would one need to prove about the following diagram to prove the Pythagorean Theorem?

Let’s see if we can do this!

Draw a line perpendicular to $\overline {AB}$ that passes though both $C$ and $\overline {A'' B''}$ . Call the intersection between this line and $\overline {AB}$ , point $E$ ; call the intersection point between this line and $\overline {A''B''}$ , point $E'$ . Explain why $\tri ACA''$ has half the area of rectangle $AEE'A''$ .

Explain why $\tri ABA'$ has half the area of square $ACC'A'$ .

Explain why $\tri ACA''$ is congruent to $\tri ABA'$ .

Explain why area of square $ACC'A'$ is equal to the area of rectangle $AEE'A''$ .

Use similar ideas to complete a proof the Pythagorean Theorem.

The converse

What is the converse to the Pythagorean Theorem? Is it true? How do you prove it?