Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter8: Complex Numbers And Polarcoordinates
Section: Chapter Questions
Problem 8CLT
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![**Problem 3: Geometry - Problem Solving**
In the diagram below of triangle \( \triangle PAO \), \( \overline{AP} \) is tangent to circle \( O \) at point \( A \). Given that \( OB = 7 \) and \( BP = 18 \), determine the length of \( AP \).
**Diagram Description:**
The diagram depicts a circle with center \( O \) and a tangent line \( \overline{AP} \) touching the circle at point \( A \). A triangle \( \triangle PAO \) is formed with the points \( P, A, \) and \( O \). Point \( B \) is on \( \overline{OP} \) such that \( OB = 7 \) and \( BP = 18 \).
Given:
- \( OB = 7 \)
- \( BP = 18 \)
Find:
- The length of \( \overline{AP} \)
**Explanation:**
1. Identify that \( OB \) and \( BP \) form a straight line segment \( OP \) with \( OB + BP = OP = 7 + 18 \).
2. Since \( \overline{AP} \) is tangent to the circle at point \( A \), \( \angle OAP \) is a right angle.
Use the Pythagorean theorem in \( \triangle OAP \):
\[ OP^2 = OB^2 + AP^2 \]
3. Calculate the total length of segment \( OP \):
\[ OP = OB + BP = 7 + 18 = 25 \]
4. Thus, \( OP = 25 \).
Use the Pythagorean theorem:
\[ OP^2 = OA^2 + AP^2 \]
\[ 25^2 = 7^2 + AP^2 \]
\[ 625 = 49 + AP^2 \]
Isolate \( AP^2 \):
\[ AP^2 = 625 - 49 \]
\[ AP^2 = 576 \]
Find \( AP \):
\[ AP = \sqrt{576} \]
\[ AP = 24 \]
Therefore, the length of \( \overline{AP} \) is \( 24 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae3ab45a-d390-4040-bd9a-ae2bbd819782%2Fc4ee730e-3a0f-44bf-bed2-b710f20aab2a%2F8ivsbmk_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Geometry - Problem Solving**
In the diagram below of triangle \( \triangle PAO \), \( \overline{AP} \) is tangent to circle \( O \) at point \( A \). Given that \( OB = 7 \) and \( BP = 18 \), determine the length of \( AP \).
**Diagram Description:**
The diagram depicts a circle with center \( O \) and a tangent line \( \overline{AP} \) touching the circle at point \( A \). A triangle \( \triangle PAO \) is formed with the points \( P, A, \) and \( O \). Point \( B \) is on \( \overline{OP} \) such that \( OB = 7 \) and \( BP = 18 \).
Given:
- \( OB = 7 \)
- \( BP = 18 \)
Find:
- The length of \( \overline{AP} \)
**Explanation:**
1. Identify that \( OB \) and \( BP \) form a straight line segment \( OP \) with \( OB + BP = OP = 7 + 18 \).
2. Since \( \overline{AP} \) is tangent to the circle at point \( A \), \( \angle OAP \) is a right angle.
Use the Pythagorean theorem in \( \triangle OAP \):
\[ OP^2 = OB^2 + AP^2 \]
3. Calculate the total length of segment \( OP \):
\[ OP = OB + BP = 7 + 18 = 25 \]
4. Thus, \( OP = 25 \).
Use the Pythagorean theorem:
\[ OP^2 = OA^2 + AP^2 \]
\[ 25^2 = 7^2 + AP^2 \]
\[ 625 = 49 + AP^2 \]
Isolate \( AP^2 \):
\[ AP^2 = 625 - 49 \]
\[ AP^2 = 576 \]
Find \( AP \):
\[ AP = \sqrt{576} \]
\[ AP = 24 \]
Therefore, the length of \( \overline{AP} \) is \( 24 \).
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