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
Concept explainers
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
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Question
![### Problem Statement
In the accompanying diagram, secant \(AB\) intersects circle \(O\) at \(D\), and secant \(AC\) intersects circle \(O\) at \(E\). Given that \(AE = 4\), \(AC = 24\), and \(AB = 16\), find \(AD\).
### Diagram Description
The given diagram illustrates a circle \(O\) with a center marked as \(O\). There are two secants drawn from point \(A\) outside the circle:
- Secant \(AB\) intersects the circle at points \(D\) and \(B\).
- Secant \(AC\) intersects the circle at points \(E\) and \(C\).
The lengths are provided as follows:
- \(AE = 4\)
- \(AC = 24\)
- \(AB = 16\)
### Analysis and Solution Approach
To solve the problem of finding the length of \(AD\), use the properties of secant segments that intersect outside the circle. According to the Secant-Secant Power Theorem, for two secants \(AB\) (where \(AD\) is part of it) and \(AC\) originating from an external point \(A\):
\[ AE \times AC = AD \times AB \]
### Step-by-Step Solution
1. Identify the given lengths in the problem:
- \(AE = 4\)
- \(AC = 24\)
- \(AB = 16\)
2. Set up the equation using the Secant-Secant Power Theorem:
\[ AE \times AC = AD \times AB \]
Substitute the given values:
\[ 4 \times 24 = AD \times 16 \]
3. Simplify and solve for \(AD\):
\[ 96 = 16 \times AD \]
\[ AD = \frac{96}{16} \]
\[ AD = 6 \]
Therefore, the length of \(AD\) is \(6\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe489ebd6-5c8e-44bd-8a5f-33d2b46a9596%2F7d1b5ed7-fcae-474f-8da2-fab93a881ce2%2Fh118726_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
In the accompanying diagram, secant \(AB\) intersects circle \(O\) at \(D\), and secant \(AC\) intersects circle \(O\) at \(E\). Given that \(AE = 4\), \(AC = 24\), and \(AB = 16\), find \(AD\).
### Diagram Description
The given diagram illustrates a circle \(O\) with a center marked as \(O\). There are two secants drawn from point \(A\) outside the circle:
- Secant \(AB\) intersects the circle at points \(D\) and \(B\).
- Secant \(AC\) intersects the circle at points \(E\) and \(C\).
The lengths are provided as follows:
- \(AE = 4\)
- \(AC = 24\)
- \(AB = 16\)
### Analysis and Solution Approach
To solve the problem of finding the length of \(AD\), use the properties of secant segments that intersect outside the circle. According to the Secant-Secant Power Theorem, for two secants \(AB\) (where \(AD\) is part of it) and \(AC\) originating from an external point \(A\):
\[ AE \times AC = AD \times AB \]
### Step-by-Step Solution
1. Identify the given lengths in the problem:
- \(AE = 4\)
- \(AC = 24\)
- \(AB = 16\)
2. Set up the equation using the Secant-Secant Power Theorem:
\[ AE \times AC = AD \times AB \]
Substitute the given values:
\[ 4 \times 24 = AD \times 16 \]
3. Simplify and solve for \(AD\):
\[ 96 = 16 \times AD \]
\[ AD = \frac{96}{16} \]
\[ AD = 6 \]
Therefore, the length of \(AD\) is \(6\).
![### Problem 4: Geometry of Circles
In the accompanying diagram of circle \(O\), secants \(CBA\) and \(CED\) intersect at \(C\). The lengths of the segments are given as follows: \(AC = 12\), \(BC = 3\), and \(DC = 9\). The goal is to find the length of segment \(EC\).
#### Diagram Description:
The image presents a circle, labeled as circle \(O\). Two secants intersect outside the circle at point \(C\). One secant extends from point \(A\) through the circle to point \(B\), continuing to point \(C\). The other secant extends from point \(D\) through the circle to point \(E\), continuing to point \(C\).
#### Given:
- \(AC = 12\)
- \(BC = 3\)
- \(DC = 9\)
#### Find:
- Length of \(EC\)
This task involves applying the properties of intersecting secants in a circle. According to the intersecting secants theorem (or the power of a point theorem), the products of the lengths of the segments of each secant are equal.
Mathematically, this can be formulated as:
\[ AC \times BC = DC \times EC \]
Use this property to find the length of \(EC\).
#### Solution Steps:
1. Calculate the product of the segments of secant \(CBA\):
\[ AC \times BC = 12 \times 3 \]
2. Set up the equation using the given lengths:
\[ 12 \times 3 = 9 \times EC \]
3. Simplify and solve for \(EC\):
\[ 36 = 9 \times EC \]
\[ EC = \frac{36}{9} \]
\[ EC = 4 \]
Thus, the length of segment \(EC\) is 4 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe489ebd6-5c8e-44bd-8a5f-33d2b46a9596%2F7d1b5ed7-fcae-474f-8da2-fab93a881ce2%2F0rzdg5v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4: Geometry of Circles
In the accompanying diagram of circle \(O\), secants \(CBA\) and \(CED\) intersect at \(C\). The lengths of the segments are given as follows: \(AC = 12\), \(BC = 3\), and \(DC = 9\). The goal is to find the length of segment \(EC\).
#### Diagram Description:
The image presents a circle, labeled as circle \(O\). Two secants intersect outside the circle at point \(C\). One secant extends from point \(A\) through the circle to point \(B\), continuing to point \(C\). The other secant extends from point \(D\) through the circle to point \(E\), continuing to point \(C\).
#### Given:
- \(AC = 12\)
- \(BC = 3\)
- \(DC = 9\)
#### Find:
- Length of \(EC\)
This task involves applying the properties of intersecting secants in a circle. According to the intersecting secants theorem (or the power of a point theorem), the products of the lengths of the segments of each secant are equal.
Mathematically, this can be formulated as:
\[ AC \times BC = DC \times EC \]
Use this property to find the length of \(EC\).
#### Solution Steps:
1. Calculate the product of the segments of secant \(CBA\):
\[ AC \times BC = 12 \times 3 \]
2. Set up the equation using the given lengths:
\[ 12 \times 3 = 9 \times EC \]
3. Simplify and solve for \(EC\):
\[ 36 = 9 \times EC \]
\[ EC = \frac{36}{9} \]
\[ EC = 4 \]
Thus, the length of segment \(EC\) is 4 units.
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