# Lesson 13

Using the Pythagorean Theorem and Similarity

### Lesson Narrative

In a subsequent lesson, students will use the altitude to the hypotenuse of a right triangle to prove the Pythagorean Theorem. An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side. This lesson sets the stage for that proof by supporting students to make sense of the similar triangles formed by drawing the altitude to the hypotenuse of a right triangle. Students also get a chance to practice using the Pythagorean Theorem, equivalent ratios, and scale factors to find unknown side lengths in similar right triangles.

Students have a chance to reason abstractly and quantitatively (MP2) as they reason about angles in the triangles formed by drawing the altitude to the hypotenuse of a right triangle using tracing paper, measurement, and logical and algebraic reasoning.

### Learning Goals

Teacher Facing

• Justify that the two smaller right triangles formed when a right triangle has an altitude drawn to its hypotenuse are similar to the original right triangle (in writing).

### Student Facing

• Let’s explore right triangles with altitudes drawn to the hypotenuse.

### Required Preparation

Ensure geometry toolkits are well stocked with tracing paper.

### Student Facing

• I can find similar triangles formed by the altitude to the hypotenuse in a right triangle.

Building On

Building Towards

### 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.