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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**Problem Statement:**
Given circle \( C \) with \( \overline{AD} \) and \( \overline{BD} \) tangent to the circle, determine \( \overline{BD} \).
Use mathematics to explain how you determined your answer.
**Diagram Explanation:**
- The circle \( C \) is depicted with a radius extending to a point \( A \) on the circumference.
- Tangent \( \overline{AD} \) touches the circle at point \( A \) and extends to point \( D \).
- Tangent \( \overline{BD} \) touches the circle at point \( B \) and extends to point \( D \).
- The lengths of the tangents are labeled as \( \overline{AD} = 6x - 2 \) and \( \overline{BD} = 4x + 8 \).
- Point \( C \) is the center of the circle.
**Solution:**
1. ***Identify Tangent Properties:***
- For a circle, tangents drawn from a common external point are equal in length.
2. ***Set up the Equation:***
\[
\overline{AD} = \overline{BD}
\]
\[
6x - 2 = 4x + 8
\]
3. ***Solve for \( x \):***
\[
6x - 4x = 8 + 2
\]
\[
2x = 10
\]
\[
x = 5
\]
4. ***Substitute \( x \) back into \( \overline{BD} \):***
\[
\overline{BD} = 4x + 8
\]
\[
\overline{BD} = 4(5) + 8
\]
\[
\overline{BD} = 20 + 8
\]
\[
\overline{BD} = 28
\]
Therefore, the length of \( \overline{BD} \) is **28 units**.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa242a625-2826-4f00-ad8f-039422d3d327%2F65fc8df3-ff09-4b00-8e25-de2f3ec63b89%2F27yj3e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Problem Statement:**
Given circle \( C \) with \( \overline{AD} \) and \( \overline{BD} \) tangent to the circle, determine \( \overline{BD} \).
Use mathematics to explain how you determined your answer.
**Diagram Explanation:**
- The circle \( C \) is depicted with a radius extending to a point \( A \) on the circumference.
- Tangent \( \overline{AD} \) touches the circle at point \( A \) and extends to point \( D \).
- Tangent \( \overline{BD} \) touches the circle at point \( B \) and extends to point \( D \).
- The lengths of the tangents are labeled as \( \overline{AD} = 6x - 2 \) and \( \overline{BD} = 4x + 8 \).
- Point \( C \) is the center of the circle.
**Solution:**
1. ***Identify Tangent Properties:***
- For a circle, tangents drawn from a common external point are equal in length.
2. ***Set up the Equation:***
\[
\overline{AD} = \overline{BD}
\]
\[
6x - 2 = 4x + 8
\]
3. ***Solve for \( x \):***
\[
6x - 4x = 8 + 2
\]
\[
2x = 10
\]
\[
x = 5
\]
4. ***Substitute \( x \) back into \( \overline{BD} \):***
\[
\overline{BD} = 4x + 8
\]
\[
\overline{BD} = 4(5) + 8
\]
\[
\overline{BD} = 20 + 8
\]
\[
\overline{BD} = 28
\]
Therefore, the length of \( \overline{BD} \) is **28 units**.
---
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