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Y 12 5 X 13 Find mZX

Y 12 5 X 13 Find mZX

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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Find m∠X

### Geometry Problem: Finding the Measure of an Angle in a Right Triangle #### Given Triangle XYZ: - **Sides:** \( XY = 5 \), \( YZ = 12 \), and \( XZ = 13 \). - **Right Angle:** \(\angle Y\) (denoted by the small square at vertex \(Y\)). #### Problem Statement: Find the measure of angle \(\angle X\). #### Answer Choices: - \( 21.04^\circ \) - \( \mathbf{22.62^\circ} \) (This option is highlighted) - \( 67.38^\circ \) - \( 68.96^\circ \) #### Solution Outline: - Since \(\angle Y\) is a right angle (90 degrees) and the triangle's sides satisfy the Pythagorean theorem (\(5^2 + 12^2 = 13^2\)), triangle \(XYZ\) is a right triangle. - Use trigonometric ratios (such as sine, cosine, or tangent) to calculate \(\angle X\). ### Graph/Diagram Explanation: - Triangle \(XYZ\) is drawn with vertices labeled as \(X\), \( Y \), and \( Z \). - \(\angle Y\) is marked as a right angle (90 degrees). - Side lengths are marked as \( XY = 5 \), \( YZ = 12 \), and \( XZ = 13 \). --- This problem and solution framework exemplify the application of trigonometric concepts in solving geometric problems, specifically within the context of right triangles.
Transcribed Image Text:### Geometry Problem: Finding the Measure of an Angle in a Right Triangle #### Given Triangle XYZ: - **Sides:** \( XY = 5 \), \( YZ = 12 \), and \( XZ = 13 \). - **Right Angle:** \(\angle Y\) (denoted by the small square at vertex \(Y\)). #### Problem Statement: Find the measure of angle \(\angle X\). #### Answer Choices: - \( 21.04^\circ \) - \( \mathbf{22.62^\circ} \) (This option is highlighted) - \( 67.38^\circ \) - \( 68.96^\circ \) #### Solution Outline: - Since \(\angle Y\) is a right angle (90 degrees) and the triangle's sides satisfy the Pythagorean theorem (\(5^2 + 12^2 = 13^2\)), triangle \(XYZ\) is a right triangle. - Use trigonometric ratios (such as sine, cosine, or tangent) to calculate \(\angle X\). ### Graph/Diagram Explanation: - Triangle \(XYZ\) is drawn with vertices labeled as \(X\), \( Y \), and \( Z \). - \(\angle Y\) is marked as a right angle (90 degrees). - Side lengths are marked as \( XY = 5 \), \( YZ = 12 \), and \( XZ = 13 \). --- This problem and solution framework exemplify the application of trigonometric concepts in solving geometric problems, specifically within the context of right triangles.
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