Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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### Trigonometry: Finding the value of x #### Problem 6: **Objective:** Solve for \( x \). #### Diagram Description: A right triangle is shown with its altitude \( \overline{AB} \) perpendicular to the base. Point \( A \) is the top vertex of the triangle, point \( B \) is the vertex where the altitude meets the base, and the third vertex (unnamed) is at the far right of the triangle. There is an angle of measure \( (3x - 15)^\circ \) at vertex \( B \). Below the triangle, it reads: \[ \overline{AB} \text{ is an altitude of the triangle} \] --- To solve the problem: 1. **Identify Known Values and Angles:** - Given angle \( \angle B = (3x - 15)^\circ \). - Since \( \overline{AB} \) is an altitude in a right triangle, one angle at vertex \( B \) is a right angle, i.e., \( 90^\circ \). 2. **Use Angle Properties:** - In a right triangle, the sum of the non-right angles must add up to 90 degrees because the total sum of angles in any triangle is 180 degrees. Therefore, for this problem: \[ (3x - 15)^\circ + 90^\circ = 180^\circ \] Simplifying this equation: \[ (3x - 15)^\circ + 90^\circ = 180^\circ \] \[ 3x - 15 = 90 \] \[ 3x = 105 \] \[ x = 35 \] Thus, the value of \( x \) is \( 35 \).
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Transcribed Image Text:### Trigonometry: Finding the value of x #### Problem 6: **Objective:** Solve for \( x \). #### Diagram Description: A right triangle is shown with its altitude \( \overline{AB} \) perpendicular to the base. Point \( A \) is the top vertex of the triangle, point \( B \) is the vertex where the altitude meets the base, and the third vertex (unnamed) is at the far right of the triangle. There is an angle of measure \( (3x - 15)^\circ \) at vertex \( B \). Below the triangle, it reads: \[ \overline{AB} \text{ is an altitude of the triangle} \] --- To solve the problem: 1. **Identify Known Values and Angles:** - Given angle \( \angle B = (3x - 15)^\circ \). - Since \( \overline{AB} \) is an altitude in a right triangle, one angle at vertex \( B \) is a right angle, i.e., \( 90^\circ \). 2. **Use Angle Properties:** - In a right triangle, the sum of the non-right angles must add up to 90 degrees because the total sum of angles in any triangle is 180 degrees. Therefore, for this problem: \[ (3x - 15)^\circ + 90^\circ = 180^\circ \] Simplifying this equation: \[ (3x - 15)^\circ + 90^\circ = 180^\circ \] \[ 3x - 15 = 90 \] \[ 3x = 105 \] \[ x = 35 \] Thus, the value of \( x \) is \( 35 \).
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