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
![### 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 \( 35 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20047395-754f-4b4f-b55f-051358973abe%2F36abc2f1-0278-4cfe-94e3-786111b8d2aa%2Fwg73av9.jpeg&w=3840&q=75)
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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