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
![**Mathematics Problem: Solving for x in a Triangle**
**Problem Statement:**
Given a triangle \( \triangle ABC \), where:
- \( AB = 8x + 5 \)
- \( CB = 10x - 1 \)
- \( BD \) is perpendicular to \( AC \)
Find the value of \( x \).
**Diagram Description:**
The diagram is a triangle \( \triangle ABC \) with the following characteristics:
- Points \( A \) and \( C \) are the base of the triangle.
- Point \( B \) is the vertex opposite the base \( AC \).
- \( BD \) is a straight line perpendicular to the base \( AC \), intersecting it at point \( D \).
- The segments \( AB \) and \( CB \) have markings indicating that they are equal in length.
- This implies that \( \triangle ABC \) is an isosceles triangle with \( AB = CB \).
**Solution Process:**
1. **Equality of Sides in an Isosceles Triangle:**
Since \( AB = CB \):
\[
8x + 5 = 10x - 1
\]
2. **Solving for \( x \):**
Simplify the equation to find the value of \( x \):
\[
8x + 5 = 10x - 1
\]
Subtract \( 8x \) from both sides:
\[
5 = 2x - 1
\]
Add 1 to both sides:
\[
6 = 2x
\]
Divide both sides by 2:
\[
x = 3
\]
**Conclusion:**
The value of \( x \) that satisfies the given conditions of the triangle is \( x = 3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c2f64fc-e07a-43b3-a4b0-6f03d401f48c%2Ffa4abe09-1cca-4b01-95d8-ae10b69598f0%2Fyv0nhj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Problem: Solving for x in a Triangle**
**Problem Statement:**
Given a triangle \( \triangle ABC \), where:
- \( AB = 8x + 5 \)
- \( CB = 10x - 1 \)
- \( BD \) is perpendicular to \( AC \)
Find the value of \( x \).
**Diagram Description:**
The diagram is a triangle \( \triangle ABC \) with the following characteristics:
- Points \( A \) and \( C \) are the base of the triangle.
- Point \( B \) is the vertex opposite the base \( AC \).
- \( BD \) is a straight line perpendicular to the base \( AC \), intersecting it at point \( D \).
- The segments \( AB \) and \( CB \) have markings indicating that they are equal in length.
- This implies that \( \triangle ABC \) is an isosceles triangle with \( AB = CB \).
**Solution Process:**
1. **Equality of Sides in an Isosceles Triangle:**
Since \( AB = CB \):
\[
8x + 5 = 10x - 1
\]
2. **Solving for \( x \):**
Simplify the equation to find the value of \( x \):
\[
8x + 5 = 10x - 1
\]
Subtract \( 8x \) from both sides:
\[
5 = 2x - 1
\]
Add 1 to both sides:
\[
6 = 2x
\]
Divide both sides by 2:
\[
x = 3
\]
**Conclusion:**
The value of \( x \) that satisfies the given conditions of the triangle is \( x = 3 \).
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