# Lesson 7

A Proof of the Pythagorean Theorem

### Lesson Narrative

In the warm-up of this lesson, students study a diagram they will use to prove the Pythagorean Theorem. In the first activity they prove the Pythagorean Theorem using the diagram. Then they apply the Pythagorean Theorem in the next activity. The final activity before the cool down is an optional look at a transformational proof of the Pythagorean Theorem.

### Learning Goals

Teacher Facing

• Calculate an unknown side length of a right triangle using the Pythagorean Theorem, and explain (orally) the reasoning.
• Explain (orally) an area-based algebraic proof of the Pythagorean Theorem.

### Student Facing

Let’s prove the Pythagorean Theorem.

### Required Preparation

If you choose to do the optional activity, you will need the 5 cut-out shapes from the Making Squares blackline master used in the first lesson of this unit—1 set of 5 for every 2 students. You will also need copies of the A Transformational Proof blackline master—1 copy for every 2 students.

### Student Facing

• I can explain why the Pythagorean Theorem is true.

Building Towards

### Glossary Entries

• Pythagorean Theorem

The Pythagorean Theorem describes the relationship between the side lengths of right triangles.

The diagram shows a right triangle with squares built on each side.  If we add the areas of the two small squares, we get the area of the larger square.

The square of the hypotenuse is equal to the sum of the squares of the legs. This is written as $$a^2+b^2=c^2$$.

• hypotenuse

The hypotenuse is the side of a right triangle that is opposite the right angle. It is the longest side of a right triangle.

Here are some right triangles. Each hypotenuse is labeled.

• legs

The legs of a right triangle are the sides that make the right angle.

Here are some right triangles. Each leg is labeled.