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In ALMN, P is a point on LM and Q is a point on LN such that PQI MN. IfLP = 4, PM = 3, and QN = 9, what is the length of LQ? Chapter St

In ALMN, P is a point on LM and Q is a point on LN such that PQI MN. IfLP = 4, PM = 3, and QN = 9, what is the length of LQ? Chapter St

Elementary Geometry For College Students, 7e
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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### 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 \
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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