Lesson 5

Working with Ratios in Right Triangles

Lesson Narrative

In a previous unit, students found unknown side lengths in right triangles using similarity. This is the first time students are expected to use trigonometric ratios rather than a specific similar triangle to find unknown side lengths in right triangles. This is not an easy concept for students, who must grapple with:

  • recognizing that the right triangle table has useful information for solving the problem
  • being given ratios within triangles, rather than being able to easily find a scale factor between two triangles
  • ratios given as a decimal value, rather than a ratio relationship like \(0.940:1\)
  • the algebraic reasoning required to find a side length that will result in a specified ratio

The goal of this lesson is for students to continue to build the understanding that an acute angle measure in a right triangle determines the ratios of its side lengths, and vice versa. Students first use their right triangle table to estimate angle measures given a right triangle with three known side lengths. Then they use the ratios they calculated to find unknown side lengths precisely. Students are not expected to master this skill in this lesson. The goal is to discover the multiple ways students may have for thinking about using the ratios they’ve computed, and to understand how students connect those ratios to the right triangles they represent (MP7).

Throughout this lesson, resist the urge to name any trigonometric ratios. Continue to refer to descriptions such as, “the ratio of the opposite leg to the adjacent leg.” If students note that the descriptions are long or wonder if there are names for these calculations, explain that mathematicians do have names for them that they'll learn soon. The purpose of continuing to use the long descriptions is so that students understand what the ratios mean and where they come from. 

Learning Goals

Teacher Facing

  • Justify an estimated angle measure using a table of ratios of side lengths of right triangles (using words and other representations).
  • Justify an estimated side length using a table of ratios of side lengths of right triangles (using words and other representations).

Student Facing

  • Let’s solve problems about right triangles.

Required Materials

Learning Targets

Student Facing

  • I can use a table of ratios of side lengths of right triangles to estimate unknown angle measures.
  • I can use a table of ratios of side lengths of right triangles to estimate unknown side lengths.

CCSS Standards

Building On

Addressing

Glossary Entries

  • complementary

    Two angles are complementary to each other if their measures add up to \(90^\circ\). The two acute angles in a right triangle are complementary to each other.