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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 \begin{align*}B\end{align*}. They will do this for four different positions of the point.

Students will discover that the ratio of \begin{align*}BC\end{align*} to \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 \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 \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 \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 \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. \begin{align*}\frac{3}{5}\end{align*}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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