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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![### Educational Website Content: Solving for \( x \) in a Tangent-Secant Circle Problem
**Problem Overview:**
We are given a geometric figure involving a circle with center \( C \). In the figure:
- \( AB \) is a secant line that intersects the circle at points \( B \) and \( D \).
- \( AD \) is a tangent to the circle at point \( D \).
- \( AB = 4x + 8 \) cm.
- \( AD = 36 \) cm.
Our goal is to solve for \( x \).
**Diagram Explanation:**
- The circle is depicted with center \( C \).
- Point \( B \) lies on the circle, making \( AB \) a secant, while point \( D \) lies on the circle where the tangent \( AD \) touches it.
- \( B \) and \( D \) are points where the line segments intersect the circle, and point \( A \) is external to the circle.
- \( AD \) (tangent) and \( AB \) (secant) create a specific relationship: the power of a point theorem (sometimes referred to as the Tangent-Secant Theorem) can be applied here.
**Using the Tangent-Secant Theorem:**
The Tangent-Secant Theorem states that:
\[ AD^2 = AB \cdot AB_{ext} \]
Where:
- \( AD \) is the length of the tangent from \( A \) to the point of tangency \( D \).
- \( AB \) is the length of the full secant segment from \( A \) through \( B \) to the circle.
- \( AB_{ext} \) is the exterior part of the secant segment from \( A \) to \( B \) (which is the same as \( AB \)).
Given:
- \( AD = 36 \) cm,
- \( AB = 4x + 8 \),
Applying the theorem:
\[ AD^2 = AB \cdot AB_{ext} \]
\[ 36^2 = (4x + 8) \cdot (4x + 8) \]
\[ 1296 = (4x + 8)^2 \]
Solve for \( x \):
1. Expand the right side:
\[ 1296 = 16x^2 + 64x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F251d803b-062a-485c-ac85-18fab7c31e4e%2F563b0a2f-b794-42df-8da4-b802a6084429%2Fbmvguea_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Website Content: Solving for \( x \) in a Tangent-Secant Circle Problem
**Problem Overview:**
We are given a geometric figure involving a circle with center \( C \). In the figure:
- \( AB \) is a secant line that intersects the circle at points \( B \) and \( D \).
- \( AD \) is a tangent to the circle at point \( D \).
- \( AB = 4x + 8 \) cm.
- \( AD = 36 \) cm.
Our goal is to solve for \( x \).
**Diagram Explanation:**
- The circle is depicted with center \( C \).
- Point \( B \) lies on the circle, making \( AB \) a secant, while point \( D \) lies on the circle where the tangent \( AD \) touches it.
- \( B \) and \( D \) are points where the line segments intersect the circle, and point \( A \) is external to the circle.
- \( AD \) (tangent) and \( AB \) (secant) create a specific relationship: the power of a point theorem (sometimes referred to as the Tangent-Secant Theorem) can be applied here.
**Using the Tangent-Secant Theorem:**
The Tangent-Secant Theorem states that:
\[ AD^2 = AB \cdot AB_{ext} \]
Where:
- \( AD \) is the length of the tangent from \( A \) to the point of tangency \( D \).
- \( AB \) is the length of the full secant segment from \( A \) through \( B \) to the circle.
- \( AB_{ext} \) is the exterior part of the secant segment from \( A \) to \( B \) (which is the same as \( AB \)).
Given:
- \( AD = 36 \) cm,
- \( AB = 4x + 8 \),
Applying the theorem:
\[ AD^2 = AB \cdot AB_{ext} \]
\[ 36^2 = (4x + 8) \cdot (4x + 8) \]
\[ 1296 = (4x + 8)^2 \]
Solve for \( x \):
1. Expand the right side:
\[ 1296 = 16x^2 + 64x
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