Using Trigonometry to Find Side Lengths of an Acute Triangle

New York State Common Core Math Geometry, Module 2, Lesson 32

Worksheets for Geometry, Module 2, Lesson 32

Student Outcomes

• Students find missing side lengths of an acute triangle given one side length and the measures of two angles.
• Students find the missing side length of an acute triangle given two side lengths and the measure of the included angle.

Using Trigonometry to Find Side Lengths of an Acute Triangle

Classwork

Opening Exercise

a. Find the lengths of 𝑑 and 𝑒.
b. Find the lengths of 𝑥 and 𝑦. How is this different from part (a)?

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Example 1

A surveyor needs to determine the distance between two points 𝐴 and 𝐵 that lie on opposite banks of a river. A point 𝐶 is chosen 160 meters from point 𝐴, on the same side of the river as 𝐴. The measures of ∠𝐵𝐴𝐶 and ∠𝐴𝐶𝐵 are 41° and 55°, respectively. Approximate the distance from 𝐴 to 𝐵 to the nearest meter.

Exercises 1–2

1. In △ 𝐴𝐵𝐶, 𝑚∠𝐴 = 30, 𝑎 = 12, and 𝑏 = 10. Find sin∠𝐵. Include a diagram in your answer.
2. A car is moving toward a tunnel carved out of the base of a hill. As the accompanying diagram shows, the top of the hill, 𝐻, is sighted from two locations, 𝐴 and 𝐵. The distance between 𝐴 and 𝐵 is 250 ft. What is the height, ℎ, of the hill to the nearest foot?

Example 2

Our friend the surveyor from Example 1 is doing some further work. He has already found the distance between points 𝐴 and 𝐵 (from Example 1). Now he wants to locate a point 𝐷 that is equidistant from both 𝐴 and 𝐵 and on the same side of the river as 𝐴. He has his assistant mark the point 𝐷 so that ∠𝐴𝐵𝐷 and ∠𝐵𝐴𝐷 both measure 75°. What is the distance between 𝐷 and 𝐴 to the nearest meter?

Exercise 3

3. Parallelogram 𝐴𝐵𝐶𝐷 has sides of lengths 44 mm and 26 mm, and one of the angles has a measure of 100°. Approximate the length of diagonal 𝐴𝐶̅̅̅̅ to the nearest millimeter.

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