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
Related questions
Question
![### Diagram Explanation
The diagram shows a circle with center \( P \) and two secants, \( \overline{AC} \) and \( \overline{CD} \), intersecting the circle at points \( B \) and \( E \) respectively. The secant \( \overline{AC} \) enters the circle at \( A \) and exits at \( C \), with intersection point \( B \) on the circumference. Secant \( \overline{CD} \) enters at \( C \), exits at \( D \), and intersects the circle at \( E \).
### Problem Statement
1. **Given:**
- \( \overline{AC} \) and \( \overline{CD} \) intersect the circle \(\odot P\) at \( B \) and \( E \) as shown.
- \( AC = 16 \)
- \( CB = 6 \)
- \( CE = 4 \)
2. **Find:**
- The length of \( CD \).
### Solution Approach
According to the secant-segment theorem, when two secants \(\overline{AC}\) and \(\overline{CD}\) intersect at a point outside the circle, the products of the lengths of the whole secant segments and their external parts are equal:
\[ (AC) \times (CB) = (CD) \times (CE) \]
### Calculations
Given:
- \( AC = 16 \), \( CB = 6 \), \( CE = 4 \)
\[ 16 \times 6 = CD \times 4 \]
\[ 96 = CD \times 4 \]
\[ CD = \frac{96}{4} \]
\[ CD = 24 \]
Thus, the length of \( CD \) is 24.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa24c9338-3a1e-453b-b98c-81e79554ce14%2F2a2347d0-e4b8-452d-8318-e5c4042e581f%2F3ezzp77_processed.png&w=3840&q=75)
Transcribed Image Text:### Diagram Explanation
The diagram shows a circle with center \( P \) and two secants, \( \overline{AC} \) and \( \overline{CD} \), intersecting the circle at points \( B \) and \( E \) respectively. The secant \( \overline{AC} \) enters the circle at \( A \) and exits at \( C \), with intersection point \( B \) on the circumference. Secant \( \overline{CD} \) enters at \( C \), exits at \( D \), and intersects the circle at \( E \).
### Problem Statement
1. **Given:**
- \( \overline{AC} \) and \( \overline{CD} \) intersect the circle \(\odot P\) at \( B \) and \( E \) as shown.
- \( AC = 16 \)
- \( CB = 6 \)
- \( CE = 4 \)
2. **Find:**
- The length of \( CD \).
### Solution Approach
According to the secant-segment theorem, when two secants \(\overline{AC}\) and \(\overline{CD}\) intersect at a point outside the circle, the products of the lengths of the whole secant segments and their external parts are equal:
\[ (AC) \times (CB) = (CD) \times (CE) \]
### Calculations
Given:
- \( AC = 16 \), \( CB = 6 \), \( CE = 4 \)
\[ 16 \times 6 = CD \times 4 \]
\[ 96 = CD \times 4 \]
\[ CD = \frac{96}{4} \]
\[ CD = 24 \]
Thus, the length of \( CD \) is 24.
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