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# 9.3: Ratios of Right Triangles

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 $B$. They will do this for four different positions of the point.

Students will discover that the ratio of $BC$ to $AB$ 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 $\triangle{ABC}$.

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 $ABC$, 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 $ABC$, 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 $ABC$, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

4. $\frac{3}{5}$

5. $\frac{4}{5}$

6. $\frac{3}{4}$

7. $\frac{4}{5}$

8. $\frac{3}{5}$

9. $\frac{4}{3}$

Position $BC$ $AB$ $\frac{BC}{AB}$ $\sin^{-1}\frac{BC}{AB}$
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 $AC$ $AB$ $\frac{AC}{AB}$ $\cos^{-1}\frac{AC}{AB}$
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. $A = \tan^{-1}\frac{BC}{AC}$

19. $A = 23.57, B = 66.42$

20. $A = 21.8, B = 68.2$

21. $A = 23.96, B = 66.04$

22. $A = 53.13, B = 36.87$

23. $A = 15.07, B = 74.93$

24. $A = 42.83, B = 47.17$

25. $A = 45, B = 45$

26. $A = 29.05, B = 60.95$

Feb 23, 2012

Nov 03, 2014