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
Related questions
Question
- \( \angle BAC = (2x + 8)^\circ \)
- \( \angle ABC = (3x - 5)^\circ \)
\( \overline{AB} \) is an altitude of the triangle.
### Explanation of Diagram
In the given diagram, there is a triangle labeled \( \triangle ABC \). The point \( A \) is the apex of the triangle, while \( B \) is located at the base middle point. Line segment \( \overline{AB} \) represents the altitude of the triangle, dropping perpendicular from \( A \) to \( B \).
The angles in the triangle are as follows:
- Angle \( BAC \) is given by \( (2x + 8)^\circ \).
- Angle \( ABC \) is given by \( (3x - 5)^\circ \).
The altitude \( \overline{AB} \) indicates that it splits the triangle into two right-angled triangles. Thus, the angles formed by \( \overline{AB} \) and the base of the triangle are right angles.
### Objective
You are required to solve for \( x \).
### Solution Approach
1. **Using Triangle Properties:** Since \( \overline{AB} \) is the altitude, it splits the triangle into two right-angled triangles, where one angle at the base \( \angle ABC \) and the other at the base opposite \( \angle BAC \) must sum up to \( 90^\circ \).
2. **Equation Setup:**
\[
(2x + 8)^\circ + (3x - 5)^\circ = 90^\circ
\]
3. **Solving the Equation:**
\[
2x + 8 + 3x - 5 = 90
\]
Simplify and solve for \( x \).
\[
5x + 3 = 90
\]
\[
5x = 87
\]
\[
x = 17.4
\]
Therefore, the value of \( x \) that satisfies the given conditions is \( 17.4 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20047395-754f-4b4f-b55f-051358973abe%2Fa82ea849-82ba-40da-93df-6b6e95904c5e%2Fh7oamo.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
#### Solve for \( x \):
10. 
- \( \angle BAC = (2x + 8)^\circ \)
- \( \angle ABC = (3x - 5)^\circ \)
\( \overline{AB} \) is an altitude of the triangle.
### Explanation of Diagram
In the given diagram, there is a triangle labeled \( \triangle ABC \). The point \( A \) is the apex of the triangle, while \( B \) is located at the base middle point. Line segment \( \overline{AB} \) represents the altitude of the triangle, dropping perpendicular from \( A \) to \( B \).
The angles in the triangle are as follows:
- Angle \( BAC \) is given by \( (2x + 8)^\circ \).
- Angle \( ABC \) is given by \( (3x - 5)^\circ \).
The altitude \( \overline{AB} \) indicates that it splits the triangle into two right-angled triangles. Thus, the angles formed by \( \overline{AB} \) and the base of the triangle are right angles.
### Objective
You are required to solve for \( x \).
### Solution Approach
1. **Using Triangle Properties:** Since \( \overline{AB} \) is the altitude, it splits the triangle into two right-angled triangles, where one angle at the base \( \angle ABC \) and the other at the base opposite \( \angle BAC \) must sum up to \( 90^\circ \).
2. **Equation Setup:**
\[
(2x + 8)^\circ + (3x - 5)^\circ = 90^\circ
\]
3. **Solving the Equation:**
\[
2x + 8 + 3x - 5 = 90
\]
Simplify and solve for \( x \).
\[
5x + 3 = 90
\]
\[
5x = 87
\]
\[
x = 17.4
\]
Therefore, the value of \( x \) that satisfies the given conditions is \( 17.4 \).
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