### Geometric Analysis of a TriangleIn 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).

Answered: Image /qna-images/answer/098c02ea-455d-4544-b056-3ecde04d1f3d.jpg

Answered: 4.5 В 3 A x + 6 E

Math

Geometry

4.5 В 3 A x + 6 E

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).

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