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
![### Understanding Secant Intersection and Solving for X
#### Diagram Explanation
The image features a geometric diagram drawn on graph paper. The diagram consists of a circle with a secant line intersecting it. Here are the detailed components of the diagram:
1. **Circle**: A circle is drawn in purple.
2. **Secant Line**: A line intersects the circle, passing through it at two points.
3. **Intersection Points**:
- The point of entry into the circle (left intersection) is marked and extends out to another point (right intersection).
- The segment of the secant line inside the circle is labeled `49`.
4. **Outer Segment of Secant**:
- The outer segment before the right intersection is labeled `6`.
- The outer segment before the left intersection is labeled `2`.
5. **Unknown Variable**: The segment of the secant line inside the circle is marked as `X`.
#### Problem Statement
Written adjacent to the circle is the instruction:
- **"Solve for x"**.
Additionally, the text "secant intersection" is written, indicating that this diagram is based on the property of the secant-secant segment theorem.
#### How to Solve
To solve for \( x \) (the unknown segment length inside the circle), we use the secant segment theorem, which states:
\[ \text{(external segment 1)} \times \text{(whole secant 1)} = \text{(external segment 2)} \times \text{(whole secant 2)} \]
Here, let \( a = 2 \) (the external segment of the left secant), \( b = 2 + x \) (the entire length of the left secant, which is the sum of the external and internal segments), \( c = 6 \) (the external segment of the right secant), and \( d = 6 + 49 = 55 \) (the entire length of the right secant, which is the sum of the external and internal segments).
According to the theorem:
\[ a \times b = c \times d \]
Substituting the values we get:
\[ 2 \times (2 + x) = 6 \times 55 \]
\[ 2(2 + x) = 330 \]
\[ 4 + 2x = 330 \]
\[ 2x =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e0b5597-0a3a-46d8-bb62-2b189b8b7954%2F152bb511-6d2d-4b2c-acb9-82b95dade7ce%2F2p4v8j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Secant Intersection and Solving for X
#### Diagram Explanation
The image features a geometric diagram drawn on graph paper. The diagram consists of a circle with a secant line intersecting it. Here are the detailed components of the diagram:
1. **Circle**: A circle is drawn in purple.
2. **Secant Line**: A line intersects the circle, passing through it at two points.
3. **Intersection Points**:
- The point of entry into the circle (left intersection) is marked and extends out to another point (right intersection).
- The segment of the secant line inside the circle is labeled `49`.
4. **Outer Segment of Secant**:
- The outer segment before the right intersection is labeled `6`.
- The outer segment before the left intersection is labeled `2`.
5. **Unknown Variable**: The segment of the secant line inside the circle is marked as `X`.
#### Problem Statement
Written adjacent to the circle is the instruction:
- **"Solve for x"**.
Additionally, the text "secant intersection" is written, indicating that this diagram is based on the property of the secant-secant segment theorem.
#### How to Solve
To solve for \( x \) (the unknown segment length inside the circle), we use the secant segment theorem, which states:
\[ \text{(external segment 1)} \times \text{(whole secant 1)} = \text{(external segment 2)} \times \text{(whole secant 2)} \]
Here, let \( a = 2 \) (the external segment of the left secant), \( b = 2 + x \) (the entire length of the left secant, which is the sum of the external and internal segments), \( c = 6 \) (the external segment of the right secant), and \( d = 6 + 49 = 55 \) (the entire length of the right secant, which is the sum of the external and internal segments).
According to the theorem:
\[ a \times b = c \times d \]
Substituting the values we get:
\[ 2 \times (2 + x) = 6 \times 55 \]
\[ 2(2 + x) = 330 \]
\[ 4 + 2x = 330 \]
\[ 2x =
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