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
![### Solving for x in a Circle with Chords
#### Problem Statement:
Find the value of \( x \).
#### Diagram Description:
The diagram consists of a circle with two intersecting chords. The chords divide the circle into four segments with the following lengths labeled:
- Segment 1: 10 units
- Segment 2: 9 units
- Segment 3: 18 units
- Segment 4: \( x - 3 \) units
#### Explanation:
To solve for \( x \), we use the property of intersecting chords in a circle, which states:
\[ (Segment 1) \times (Segment 2) = (Segment 3) \times (Segment 4) \]
Given the lengths, we substitute these values into the property:
\[ 10 \times 9 = 18 \times (x - 3) \]
### Steps to Solve:
1. Calculate the left-hand side:
\[ 10 \times 9 = 90 \]
2. Set up the equation:
\[ 90 = 18 \times (x - 3) \]
3. Expand the right-hand side:
\[ 90 = 18x - 54 \]
4. Isolate \( x \) by adding 54 to both sides:
\[ 90 + 54 = 18x \]
\[ 144 = 18x \]
5. Solve for \( x \):
\[ x = \frac{144}{18} \]
\[ x = 8 \]
Thus, the value of \( x \) is \( 8 \).
#### Solution:
\[ x = 8 \]
This step-by-step explanation helps in understanding how to use the property of intersecting chords to solve for an unknown segment length in a circle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0529fcd2-ed2d-4329-a507-c02ab4416f64%2F48aa6d15-f12f-4d27-bd93-f90606709072%2F070kjc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving for x in a Circle with Chords
#### Problem Statement:
Find the value of \( x \).
#### Diagram Description:
The diagram consists of a circle with two intersecting chords. The chords divide the circle into four segments with the following lengths labeled:
- Segment 1: 10 units
- Segment 2: 9 units
- Segment 3: 18 units
- Segment 4: \( x - 3 \) units
#### Explanation:
To solve for \( x \), we use the property of intersecting chords in a circle, which states:
\[ (Segment 1) \times (Segment 2) = (Segment 3) \times (Segment 4) \]
Given the lengths, we substitute these values into the property:
\[ 10 \times 9 = 18 \times (x - 3) \]
### Steps to Solve:
1. Calculate the left-hand side:
\[ 10 \times 9 = 90 \]
2. Set up the equation:
\[ 90 = 18 \times (x - 3) \]
3. Expand the right-hand side:
\[ 90 = 18x - 54 \]
4. Isolate \( x \) by adding 54 to both sides:
\[ 90 + 54 = 18x \]
\[ 144 = 18x \]
5. Solve for \( x \):
\[ x = \frac{144}{18} \]
\[ x = 8 \]
Thus, the value of \( x \) is \( 8 \).
#### Solution:
\[ x = 8 \]
This step-by-step explanation helps in understanding how to use the property of intersecting chords to solve for an unknown segment length in a circle.
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