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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Help me please I don't understand my hw
![### Problem Statement and Diagram Explanation
**Problem:**
1. In the diagram, **\(\overline{AB}\)** is a tangent and **\(\overline{BD}\)** is a secant with \(AB = 6\) and \(CD = 5\). Find \(BC\).
**Diagram Explanation:**
The given diagram consists of a circle with the following line segments:
- **\(\overline{AB}\)** is a tangent to the circle, meaning it touches the circle at exactly one point.
- **\(\overline{BD}\)** is a secant, meaning it intersects the circle at two points, \(C\) and \(D\).
- Points \(A\) and \(B\) are on the tangent line outside the circle.
- Point \(A\) is where the tangent touches the circle.
- Points \(C\) and \(D\) are where the secant intersects the circle, with point \(C\) being closer to point \(B\).
Given:
- Length of the tangent \(AB = 6\)
- Length \(CD = 5\)
The goal is to find the length of \(BC\).
**Notes:**
- Since \(BD\) is a secant, the diagram implies a relationship between the lengths of specific segments using properties of tangents and secants.
**Calculation Method:**
Utilization of the Secant-Tangent Theorem could be applied here, which states that the tangent squared is equal to the product of the external part of the secant segment and the whole secant segment.
\[ \overline{AB}^2 = \overline{BC} \cdot \overline{BD} \]
Given values can then be substituted into this theorem to solve for \(BC\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f655a5e-3946-44a0-87d7-d06571787362%2F30461f94-a005-4c80-959a-04b195c525a2%2Fulpnpl5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement and Diagram Explanation
**Problem:**
1. In the diagram, **\(\overline{AB}\)** is a tangent and **\(\overline{BD}\)** is a secant with \(AB = 6\) and \(CD = 5\). Find \(BC\).
**Diagram Explanation:**
The given diagram consists of a circle with the following line segments:
- **\(\overline{AB}\)** is a tangent to the circle, meaning it touches the circle at exactly one point.
- **\(\overline{BD}\)** is a secant, meaning it intersects the circle at two points, \(C\) and \(D\).
- Points \(A\) and \(B\) are on the tangent line outside the circle.
- Point \(A\) is where the tangent touches the circle.
- Points \(C\) and \(D\) are where the secant intersects the circle, with point \(C\) being closer to point \(B\).
Given:
- Length of the tangent \(AB = 6\)
- Length \(CD = 5\)
The goal is to find the length of \(BC\).
**Notes:**
- Since \(BD\) is a secant, the diagram implies a relationship between the lengths of specific segments using properties of tangents and secants.
**Calculation Method:**
Utilization of the Secant-Tangent Theorem could be applied here, which states that the tangent squared is equal to the product of the external part of the secant segment and the whole secant segment.
\[ \overline{AB}^2 = \overline{BC} \cdot \overline{BD} \]
Given values can then be substituted into this theorem to solve for \(BC\).
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