# Proof of the Pythagorean Theorem

Video solutions to help Grade 8 students learn how to apply the Pythagorean Theorem to find lengths of right triangles in two dimensions.

## New York State Common Core Math Module 3, Grade 8, Lesson 13

### Lesson 13 Student Outcomes

• Students practice applying the Pythagorean Theorem to find lengths of right triangles in two dimensions.
• We have to determine whether or not we actually have enough information to use properties of similar triangles to solve problems.

### NYS Math Module 3 Grade 8 Lesson 13

Classwork

Discussion
The following proof of the Pythagorean Theorem is based on the fact that similarity is transitive. It begins with the right triangle, shown on the next page, split into two other right triangles. The three triangles are placed in the same orientation, and students verify that one pair of triangles are similar using the AA criterion, then a second pair of triangles are shown to be similar using the AA criterion, and then finally all three triangles are shown to be similar by the fact that similarity is transitive. Once it is shown that all three triangles are in fact similar, the theorem is proved by comparing the ratios of corresponding side lengths. Because some of the triangles share side lengths that are the same (or sums of lengths), then the formula a2 + b2 = c2 is derived.

Exercises
Use the Pythagorean Theorem to determine the unknown length of the right triangle.
1. Determine the length of side c in each of the triangles below.

2. Determine the length of side b in each of the triangles below.

3. Determine the length of QS. (Hint: Use the Pythagorean Theorem twice.)

The following video shows the Pythagorean Theorem proof using similar triangles: