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Elementary Geometry For College Students, 7e
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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**Parallelogram Problem Solution** **Problem Statement:** 26. Find \( AM \) in the parallelogram if \( PN = 9 \) and \( AO = 4 \). **Diagram Explanation:** The given image depicts a parallelogram \( APON \) with diagonals \( PA \) and \( ON \) intersecting at point \( M \). The points \( A \), \( P \), \( O \), and \( N \) are vertices of the parallelogram. Given values are \( PN = 9 \) and \( AO = 4 \). **Solution:** 1. **Understand the Properties of a Parallelogram:** - In a parallelogram, the diagonals bisect each other. - Therefore, \( M \) is the midpoint of both diagonals \( PA \) and \( ON \). 2. **Use the Bisection Property:** - Since \( PN \) is a diagonal and \( M \) is its midpoint, \( PM = \frac{PN}{2} = \frac{9}{2} = 4.5 \). - Similarly, since \( AO \) is a diagonal and \( M \) is its midpoint, \( AM = \frac{AO}{2} = \frac{4}{2} = 2 \). **Answer:** Therefore, in the given parallelogram, the length of \( AM \) is \( 2 \) units. **Summary:** By utilizing the properties of the diagonals in a parallelogram, we determined that the length of \( AM \) is \( 2 \) units, given that \( PN = 9 \) and \( AO = 4 \). The key step was recognizing that the diagonals bisect each other, thus allowing us to find the required segment lengths easily.
Transcribed Image Text:**Parallelogram Problem Solution** **Problem Statement:** 26. Find \( AM \) in the parallelogram if \( PN = 9 \) and \( AO = 4 \). **Diagram Explanation:** The given image depicts a parallelogram \( APON \) with diagonals \( PA \) and \( ON \) intersecting at point \( M \). The points \( A \), \( P \), \( O \), and \( N \) are vertices of the parallelogram. Given values are \( PN = 9 \) and \( AO = 4 \). **Solution:** 1. **Understand the Properties of a Parallelogram:** - In a parallelogram, the diagonals bisect each other. - Therefore, \( M \) is the midpoint of both diagonals \( PA \) and \( ON \). 2. **Use the Bisection Property:** - Since \( PN \) is a diagonal and \( M \) is its midpoint, \( PM = \frac{PN}{2} = \frac{9}{2} = 4.5 \). - Similarly, since \( AO \) is a diagonal and \( M \) is its midpoint, \( AM = \frac{AO}{2} = \frac{4}{2} = 2 \). **Answer:** Therefore, in the given parallelogram, the length of \( AM \) is \( 2 \) units. **Summary:** By utilizing the properties of the diagonals in a parallelogram, we determined that the length of \( AM \) is \( 2 \) units, given that \( PN = 9 \) and \( AO = 4 \). The key step was recognizing that the diagonals bisect each other, thus allowing us to find the required segment lengths easily.
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