Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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![### Geometry Problem: Finding Lengths in Parallel Segments
**Problem Statement:**
In triangle \( \Delta LMN \), P is a point on segment \( \overline{LM} \) and Q is a point on segment \( \overline{LN} \) such that \( \overline{PQ} \) is parallel to \( \overline{MN} \). If \( LP = 4 \), \( PM = 3 \), and \( QN = 9 \), what is the length of \( LQ \)?
**Given:**
1. \( \Delta LMN \)
2. \( P \) is a point on \( \overline{LM} \)
3. \( Q \) is a point on \( \overline{LN} \)
4. \( \overline{PQ} \parallel \overline{MN} \)
5. \( LP = 4 \)
6. \( PM = 3 \)
7. \( QN = 9 \)
**Find:**
- The length of \( LQ \).
**Important Concepts:**
- **Parallel Lines and Proportional Segments:** When a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally.
---
**Solution Steps:**
1. Identify the lengths:
\[
LM = LP + PM = 4 + 3 = 7
\]
2. Use the proportionality theorem:
\[
\frac{LP}{LM} = \frac{LQ}{LN}
\]
Given \( QN = 9 \),
\[
\text{Then, } LN = LQ + QN
\]
3. Let \( LQ = x \). Hence,
\[
LN = x + 9
\]
4. Now, substitute the known values into the proportional equation:
\[
\frac{4}{7} = \frac{x}{x + 9}
\]
5. Cross-multiply to solve for \( x \):
\[
4(x + 9) = 7x
\]
\[
4x + 36 = 7x
\]
\[
36 = 3x
\]
\[
x = 12
\](https://content.bartleby.com/qna-images/question/3e7bbe56-6e0f-4e60-be65-bd82ac2b3970/ca9a9124-288e-4b8f-b3a4-9ee5e78b5be6/kbq8949_processed.jpeg)
Transcribed Image Text:### Geometry Problem: Finding Lengths in Parallel Segments
**Problem Statement:**
In triangle \( \Delta LMN \), P is a point on segment \( \overline{LM} \) and Q is a point on segment \( \overline{LN} \) such that \( \overline{PQ} \) is parallel to \( \overline{MN} \). If \( LP = 4 \), \( PM = 3 \), and \( QN = 9 \), what is the length of \( LQ \)?
**Given:**
1. \( \Delta LMN \)
2. \( P \) is a point on \( \overline{LM} \)
3. \( Q \) is a point on \( \overline{LN} \)
4. \( \overline{PQ} \parallel \overline{MN} \)
5. \( LP = 4 \)
6. \( PM = 3 \)
7. \( QN = 9 \)
**Find:**
- The length of \( LQ \).
**Important Concepts:**
- **Parallel Lines and Proportional Segments:** When a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally.
---
**Solution Steps:**
1. Identify the lengths:
\[
LM = LP + PM = 4 + 3 = 7
\]
2. Use the proportionality theorem:
\[
\frac{LP}{LM} = \frac{LQ}{LN}
\]
Given \( QN = 9 \),
\[
\text{Then, } LN = LQ + QN
\]
3. Let \( LQ = x \). Hence,
\[
LN = x + 9
\]
4. Now, substitute the known values into the proportional equation:
\[
\frac{4}{7} = \frac{x}{x + 9}
\]
5. Cross-multiply to solve for \( x \):
\[
4(x + 9) = 7x
\]
\[
4x + 36 = 7x
\]
\[
36 = 3x
\]
\[
x = 12
\
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