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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Question
![### Geometric Analysis of a Triangle
In the following diagram, we have a triangle with vertices labeled \( A \), \( B \), \( C \), \( D \), and \( E \).
1. **Length of segments:**
- Segment \( AB = 3 \)
- Segment \( BC = 4.5 \)
- Segment \( BE = x \)
- Segment \( ED = x + 6 \)
2. **Angles and Perpendiculars:**
- Line \( BE \) is perpendicular to line \( AD \)
- Line \( ED \) is perpendicular to line \( AD \)
3. **Question:**
- Find the length of segment \( CD \).
This geometrical setting presents a triangle with internal perpendicular parts contributing to additional smaller perpendicular triangles within the main triangle. \( BE \) and \( ED \) denote heights of the smaller triangles, implying congruency in the triangle segment relationships.
**Steps to Solve for \( CD \):**
Using the understanding of triangle properties and the Pythagorean theorem, we need to calculate the lengths of segments using algebraic expressions and solve for \( x \) to find \( CD \).
Since the triangles \( \triangle ABE \) and \( \triangle DEC \) have perpendiculars extending from shared heights \( x \) and \( x + 6 \):
1. Set up proportion:
\[
\frac{AB}{AE} = \frac{BC}{CD}
\]
2. Plug in given lengths and solve for \( CD \):
\[
\frac{3}{4.5} = \frac{x}{x+6}
\]
By solving the resulting equations, you should be able to find the length of \( CD \).
**Answer:** The length of \( CD = 10 \) (This can be arrived by solving the equation step by step as shown).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb73d9f23-fb52-4407-b592-aeda313b8dae%2F098c02ea-455d-4544-b056-3ecde04d1f3d%2Fv4zs68i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometric Analysis of a Triangle
In the following diagram, we have a triangle with vertices labeled \( A \), \( B \), \( C \), \( D \), and \( E \).
1. **Length of segments:**
- Segment \( AB = 3 \)
- Segment \( BC = 4.5 \)
- Segment \( BE = x \)
- Segment \( ED = x + 6 \)
2. **Angles and Perpendiculars:**
- Line \( BE \) is perpendicular to line \( AD \)
- Line \( ED \) is perpendicular to line \( AD \)
3. **Question:**
- Find the length of segment \( CD \).
This geometrical setting presents a triangle with internal perpendicular parts contributing to additional smaller perpendicular triangles within the main triangle. \( BE \) and \( ED \) denote heights of the smaller triangles, implying congruency in the triangle segment relationships.
**Steps to Solve for \( CD \):**
Using the understanding of triangle properties and the Pythagorean theorem, we need to calculate the lengths of segments using algebraic expressions and solve for \( x \) to find \( CD \).
Since the triangles \( \triangle ABE \) and \( \triangle DEC \) have perpendiculars extending from shared heights \( x \) and \( x + 6 \):
1. Set up proportion:
\[
\frac{AB}{AE} = \frac{BC}{CD}
\]
2. Plug in given lengths and solve for \( CD \):
\[
\frac{3}{4.5} = \frac{x}{x+6}
\]
By solving the resulting equations, you should be able to find the length of \( CD \).
**Answer:** The length of \( CD = 10 \) (This can be arrived by solving the equation step by step as shown).
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