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# 9.3: Ratios of Right Triangles

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 8, Lesson 5.

ID: 11576

Time Required: 45 minutes

## Activity Overview

In this activity, students will explore the ratios of right triangles. Students will discover that they can find the measure of the angles of a right triangle given the length of any two sides.

Topic: Right Triangles & Trigonometric Ratios

• Sine
• Cosine
• Tangent

Teacher Preparation and Notes

Associated Materials

## Problem 1 – Exploring Right Triangle Trigonometry

You may need to allow students to use a textbook (or other resource) to find the definitions of sine, cosine, and tangent.

Students are asked to give the ratio of several triangles on their handheld or their accompanying worksheet.

## Problem 2 – Exploring the Sine Ratio of a Right Triangle

For this problem, students will investigate the sine ratio of two sides of a triangle. Students should start the Cabri Jr. app and open the file Trig.8xv.

Students will collect data on their worksheets by moving point B\begin{align*}B\end{align*}. They will do this for four different positions of the point.

Students will discover that the ratio of BC\begin{align*}BC\end{align*} to AB\begin{align*}AB\end{align*} remains constant, no matter how large the triangle is; Therefore, students will be able to use the inverse of sine to find the measure of the angles in ABC\begin{align*}\triangle{ABC}\end{align*}.

Students will need to answer several questions on their handhelds or their accompanying worksheets.

## Problem 3 – Exploring the Cosine Ratio of a Right Triangle

Students will repeat the exploration in Problem 2, but with the cosine ratio.

## Problem 4 – Applying the Sine, Cosine, and Tangent Ratio of a Right Triangle

In Problem 4, students are asked to apply what they have learned about how to find the measure of an angle of a right triangle given two sides of the triangle.

## Solutions

1. For right triangle ABC\begin{align*}ABC\end{align*}, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

2. For right triangle ABC\begin{align*}ABC\end{align*}, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

3. For right triangle ABC\begin{align*}ABC\end{align*}, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

4. 35\begin{align*}\frac{3}{5}\end{align*}

5. 45\begin{align*}\frac{4}{5}\end{align*}

6. 34\begin{align*}\frac{3}{4}\end{align*}

7. 45\begin{align*}\frac{4}{5}\end{align*}

8. 35\begin{align*}\frac{3}{5}\end{align*}

9. 43\begin{align*}\frac{4}{3}\end{align*}

Position BC\begin{align*}BC\end{align*} AB\begin{align*}AB\end{align*} BCAB\begin{align*}\frac{BC}{AB}\end{align*} sin1BCAB\begin{align*}\sin^{-1}\frac{BC}{AB}\end{align*}
1 2.4376781463393 6.2006451991814 0.39313298342907 23.149583787224
2 3.0769811671077 7.8268201774092 0.39313298342907 23.149583787224
3 3.6665092204124 9.3263841370716 0.39313298342906 23.149583787223
4 4.3154341679767 10.977034107736 0.39313298342905 23.149583787222

11. The ratio does not change.

12. No, the angle does not change.

13. 23.1496

14. 66.8504

Position AC\begin{align*}AC\end{align*} AB\begin{align*}AB\end{align*} ACAB\begin{align*}\frac{AC}{AB}\end{align*} cos1ACAB\begin{align*}\cos^{-1}\frac{AC}{AB}\end{align*}
1 7.0816099136391 8.8549125969341 0.7997379800328 36.894911430193
2 8.0238624186986 10.033114118664 0.79973798003277 36.894911430196
3 9.0235078139592 11.283080257848 0.79973798003279 36.894911430194
4 3.7704816328074 4.7146462053143 0.79973798003281 36.894911430192

16. 36.8949

17. 53.1051

18. A=tan1BCAC\begin{align*}A = \tan^{-1}\frac{BC}{AC}\end{align*}

19. A=23.57,B=66.42\begin{align*}A = 23.57, B = 66.42\end{align*}

20. A=21.8,B=68.2\begin{align*}A = 21.8, B = 68.2\end{align*}

21. A=23.96,B=66.04\begin{align*}A = 23.96, B = 66.04\end{align*}

22. A=53.13,B=36.87\begin{align*}A = 53.13, B = 36.87\end{align*}

23. A=15.07,B=74.93\begin{align*}A = 15.07, B = 74.93\end{align*}

24. A=42.83,B=47.17\begin{align*}A = 42.83, B = 47.17\end{align*}

25. A=45,B=45\begin{align*}A = 45, B = 45\end{align*}

26. A=29.05,B=60.95\begin{align*}A = 29.05, B = 60.95\end{align*}

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