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83 O O O O Question 12 Find the missing Side lengihs. 8J3 3 n=4 M= 4, n=8 m=8,n=4 m= 8,n=2 30 43

83 O O O O Question 12 Find the missing Side lengihs. 8J3 3 n=4 M= 4, n=8 m=8,n=4 m= 8,n=2 30 43

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Question 12: Find the Missing Side Lengths In this geometry problem, we are given a right triangle with one angle measuring 30°, one side labeled \( 4\sqrt{3} \), and two unknown side lengths labeled as \( m \) and \( n \). Here is the diagram of the triangle presented in the problem: ``` n /| / | / | / | m 30° /____| ``` The given side length is \( 4\sqrt{3} \), opposite to the 30° angle. ### Solution Steps: 1. **Identify the sides in terms of known values and trigonometric ratios:** For a 30°-60°-90° triangle: - The hypotenuse (m) is twice the length of the shorter leg. - The side opposite the 30° angle (n) is half the hypotenuse. - The longer leg (\( n \)), opposite the 60° angle, is \( \sqrt{3} \) times the shorter leg. 2. **Given:** - Opposite the 30° angle: \( 4\sqrt{3} \) - Use trigonometric relationships to find \( m \) and \( n \). 3. **Calculate all sides:** - Since \( 4\sqrt{3} \) is the side opposite the 60° angle, we know that it is \( \sqrt{3} \) times the length of the shorter side. - Let the shorter side be \( a \). Formula: \( a\sqrt{3} = 4\sqrt{3} \) Solve for \( a \): \( a = 4 \) - Hence, the shorter side (n): \( n = a = 4 \) - And the hypotenuse \( m \): \( m = 2a = 2 \times 4 = 8 \) ### Final Calculations (Repeated for verification in the given problem): 1. \( m = 4 \sqrt{3} \) 2. Given: \( n = 4 \) 3. Recalculated: - \( m = 8 \) - \( n = 4 \) In conclusion, the values of the missing side lengths of the triangle are \( m = 8 \
Transcribed Image Text:### Question 12: Find the Missing Side Lengths In this geometry problem, we are given a right triangle with one angle measuring 30°, one side labeled \( 4\sqrt{3} \), and two unknown side lengths labeled as \( m \) and \( n \). Here is the diagram of the triangle presented in the problem: ``` n /| / | / | / | m 30° /____| ``` The given side length is \( 4\sqrt{3} \), opposite to the 30° angle. ### Solution Steps: 1. **Identify the sides in terms of known values and trigonometric ratios:** For a 30°-60°-90° triangle: - The hypotenuse (m) is twice the length of the shorter leg. - The side opposite the 30° angle (n) is half the hypotenuse. - The longer leg (\( n \)), opposite the 60° angle, is \( \sqrt{3} \) times the shorter leg. 2. **Given:** - Opposite the 30° angle: \( 4\sqrt{3} \) - Use trigonometric relationships to find \( m \) and \( n \). 3. **Calculate all sides:** - Since \( 4\sqrt{3} \) is the side opposite the 60° angle, we know that it is \( \sqrt{3} \) times the length of the shorter side. - Let the shorter side be \( a \). Formula: \( a\sqrt{3} = 4\sqrt{3} \) Solve for \( a \): \( a = 4 \) - Hence, the shorter side (n): \( n = a = 4 \) - And the hypotenuse \( m \): \( m = 2a = 2 \times 4 = 8 \) ### Final Calculations (Repeated for verification in the given problem): 1. \( m = 4 \sqrt{3} \) 2. Given: \( n = 4 \) 3. Recalculated: - \( m = 8 \) - \( n = 4 \) In conclusion, the values of the missing side lengths of the triangle are \( m = 8 \
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