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
Topic Video
Question
![### Geometry Problem: Finding the Length of VW
**Problem Statement:**
Given triangles \( \Delta VXY \) and \( \Delta WYZ \), what is the length of \( VW \)?
**Diagram Explanation:**
The diagram shows two right triangles, \( \Delta VXY \) and \( \Delta WYZ \), which share angle \( Y \). Here are the key elements in the diagram:
- \( VXY \) is a right triangle with a right angle at \( Y \).
- \( Z \) is a point on \( XY \) such that \( WZ \) is perpendicular to \( XY \).
- The hypotenuse \( VX \) measures 62.5 units.
- Side \( XY \) is divided into two segments by point \( Z \): \( XZ \) (not explicitly labeled with a length) and \( ZY \), measuring 50 units.
- The total length \( VY \) is given as 80 units.
**Task:**
Calculate the length of segment \( VW \).
**Answer Options:**
1. 37.5
2. 30.25
3. 104.2
4. 10.75
### Solution:
Let's consider using geometric properties and proportions to find the length \( VW \).
Given:
- \( VX = 62.5 \)
- \( XY = 50 \)
- \( VY = 80 \)
First, recognize that in right triangles sharing a similar angle, the ratios of corresponding sides are equal.
In right triangle \( \Delta VXY \):
- \( VY = 80 \) is the hypotenuse, and \( XY = 50 \).
Using the Pythagorean theorem:
\[ VY^2 = VX^2 + XY^2 \]
\[ 80^2 = 62.5^2 + XY^2 \]
\[ 6400 = 3906.25 + XY^2 \]
\[ XY^2 = 6400 - 3906.25 \]
\[ XY^2 = 2493.75 \]
\[ XY = \sqrt{2493.75} \]
Taking square roots:
\[ XY \approx 49.94 \]
Since (\( \approx 50) is only slightly longer.
Given \( XY = 50 \):
- \( WY \) in \( \Delta WYZ \) is the segment \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb294f3d7-c37e-4d19-872e-5bb2e2baac27%2Fbd0669ce-0c07-41bf-8b7a-ed9cce366e6b%2Fi5y4jad_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Finding the Length of VW
**Problem Statement:**
Given triangles \( \Delta VXY \) and \( \Delta WYZ \), what is the length of \( VW \)?
**Diagram Explanation:**
The diagram shows two right triangles, \( \Delta VXY \) and \( \Delta WYZ \), which share angle \( Y \). Here are the key elements in the diagram:
- \( VXY \) is a right triangle with a right angle at \( Y \).
- \( Z \) is a point on \( XY \) such that \( WZ \) is perpendicular to \( XY \).
- The hypotenuse \( VX \) measures 62.5 units.
- Side \( XY \) is divided into two segments by point \( Z \): \( XZ \) (not explicitly labeled with a length) and \( ZY \), measuring 50 units.
- The total length \( VY \) is given as 80 units.
**Task:**
Calculate the length of segment \( VW \).
**Answer Options:**
1. 37.5
2. 30.25
3. 104.2
4. 10.75
### Solution:
Let's consider using geometric properties and proportions to find the length \( VW \).
Given:
- \( VX = 62.5 \)
- \( XY = 50 \)
- \( VY = 80 \)
First, recognize that in right triangles sharing a similar angle, the ratios of corresponding sides are equal.
In right triangle \( \Delta VXY \):
- \( VY = 80 \) is the hypotenuse, and \( XY = 50 \).
Using the Pythagorean theorem:
\[ VY^2 = VX^2 + XY^2 \]
\[ 80^2 = 62.5^2 + XY^2 \]
\[ 6400 = 3906.25 + XY^2 \]
\[ XY^2 = 6400 - 3906.25 \]
\[ XY^2 = 2493.75 \]
\[ XY = \sqrt{2493.75} \]
Taking square roots:
\[ XY \approx 49.94 \]
Since (\( \approx 50) is only slightly longer.
Given \( XY = 50 \):
- \( WY \) in \( \Delta WYZ \) is the segment \(
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning