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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![**Problem Statement:**
Find the value of \( x \) and \( y \) if \( \triangle LUV \sim \triangle GEO \).
**Diagrams and Explanation:**
1. **Triangle \( \triangle LUV \):**
- Side \( LU = 6 \)
- Side \( UV = 10 \)
- Side \( LV = x \)
2. **Triangle \( \triangle GEO \):**
- Side \( GE = 4 \)
- Side \( EO = y \)
- Side \( GO = 8 \)
**Concept Applied:**
Since the triangles \( \triangle LUV \) and \( \triangle GEO \) are similar, the corresponding sides are proportional. Thus, the following proportion can be established:
\[
\frac{LU}{GE} = \frac{UV}{EO} = \frac{LV}{GO}
\]
Using the given side lengths:
\[
\frac{6}{4} = \frac{10}{y} = \frac{x}{8}
\]
**Solution:**
1. To find \( y \), use the proportion \(\frac{6}{4} = \frac{10}{y}\):
\[
\frac{3}{2} = \frac{10}{y}
\]
\[
3y = 20
\]
\[
y = \frac{20}{3}
\]
2. To find \( x \), use the proportion \(\frac{6}{4} = \frac{x}{8}\):
\[
\frac{3}{2} = \frac{x}{8}
\]
\[
3 \cdot 8 = 2x
\]
\[
24 = 2x
\]
\[
x = 12
\]
**Conclusion:**
The values of the unknown sides are:
- \( x = 12 \)
- \( y = \frac{20}{3} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F830af2fd-c9a7-4fc7-9b97-f56029949fa9%2F4c5423e8-0456-455e-be14-9e89de1b7fe2%2F5txa97k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the value of \( x \) and \( y \) if \( \triangle LUV \sim \triangle GEO \).
**Diagrams and Explanation:**
1. **Triangle \( \triangle LUV \):**
- Side \( LU = 6 \)
- Side \( UV = 10 \)
- Side \( LV = x \)
2. **Triangle \( \triangle GEO \):**
- Side \( GE = 4 \)
- Side \( EO = y \)
- Side \( GO = 8 \)
**Concept Applied:**
Since the triangles \( \triangle LUV \) and \( \triangle GEO \) are similar, the corresponding sides are proportional. Thus, the following proportion can be established:
\[
\frac{LU}{GE} = \frac{UV}{EO} = \frac{LV}{GO}
\]
Using the given side lengths:
\[
\frac{6}{4} = \frac{10}{y} = \frac{x}{8}
\]
**Solution:**
1. To find \( y \), use the proportion \(\frac{6}{4} = \frac{10}{y}\):
\[
\frac{3}{2} = \frac{10}{y}
\]
\[
3y = 20
\]
\[
y = \frac{20}{3}
\]
2. To find \( x \), use the proportion \(\frac{6}{4} = \frac{x}{8}\):
\[
\frac{3}{2} = \frac{x}{8}
\]
\[
3 \cdot 8 = 2x
\]
\[
24 = 2x
\]
\[
x = 12
\]
**Conclusion:**
The values of the unknown sides are:
- \( x = 12 \)
- \( y = \frac{20}{3} \)
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