Between-Figure and Within-Figure Ratios

New York State Common Core Math Geometry, Module 2, Lesson 16

Worksheets for Geometry

Student Outcomes

• Students indirectly solve for measurements involving right triangles using scale factors, ratios between similar figures, and ratios within similar figures.
• Students use trigonometric ratios to solve applied problems.

Between-Figure and Within-Figure Ratios

Classwork

Opening Exercise

At a certain time of day, a 12 m flagpole casts an 8 m shadow. Write an equation that would allow you to find the height, ℎ, of the tree that uses the length, 𝑠, of the tree’s shadow.

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Example 1

Given △ 𝐴𝐵𝐶 ~ △ 𝐴′𝐵′𝐶′, find the missing side lengths.

Example 2

In the diagram above, a large flagpole stands outside of an office building. Marquis realizes that when he looks up from the ground 60 m away from the flagpole, the top of the flagpole and the top of the building line up. If the flagpole is 35 m tall and Marquis is 170 m from the building, how tall is the building? a. Are the triangles in the diagram similar? Explain.
b. Determine the height of the building using what you know about scale factors.
c. Determine the height of the building using ratios between similar figures.
d. Determine the height of the building using ratios within similar figures.

Example 3

The following right triangles are similar. We will determine the unknown side lengths by using ratios within the first triangle. For each of the triangles below, we define the base as the horizontal length of the triangle and the height as the vertical length.
a. Write and find the value of the ratio that compares the height to the hypotenuse of the leftmost triangle.
b. Write and find the value of the ratio that compares the base to the hypotenuse of the leftmost triangle.
c. Write and find the value of the ratio that compares the height to the base of the leftmost triangle.
d. Use the triangle with lengths 3– 4– 5 and triangle 𝐴 to answer the following questions:
i. Which ratio can be used to determine the height of triangle 𝐴?
ii. Which ratio can be used to determine the hypotenuse of triangle 𝐴?
iii. Find the unknown lengths of triangle 𝐴.
e. Use the triangle with lengths 3– 4– 5 and triangle 𝐵 to answer the following questions:
i. Which ratio can be used to determine the base of triangle 𝐵?
ii. Which ratio can be used to determine the hypotenuse of triangle 𝐵?
iii. Find the unknown lengths of triangle 𝐵.
f. Use the triangle with lengths 3– 4– 5 and triangle 𝐶 to answer the following questions:
i. Which ratio can be used to determine the height of triangle 𝐶?
ii. Which ratio can be used to determine the base of triangle 𝐶?
iii. Find the unknown lengths of triangle 𝐶.
g. Explain the relationship of the ratio of the corresponding sides within a figure to the ratio of the corresponding sides within a similar figure.
h. How does the relationship you noted in part (g) allow you to determine the length of an unknown side of a triangle?

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