### 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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Answered: AB is an altitude of the triangle

Math

Geometry

AB is an altitude of the triangle

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 $.

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