9.3: Ratios of Right Triangles
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
- This activity was written to be explored on the TI-84 with the Cabri Jr. and Learning Check applications.
- Before beginning this activity, make sure that all students have the Cabri Jr. applications. Also, make sure students have or know the trigonometric definitions.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
- To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=11576 and select TRIG.
Associated Materials
- Student Worksheet: Ratios of Right Triangles http://www.ck12.org/flexr/chapter/9693, scroll down to the third activity.
- Cabri Jr. Application
- TRIG.8xv
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*}
10. Sample answers:
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
15. Sample answers:
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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