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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![**Identifying Dilation in a Graph**
In the given problem, we are tasked with identifying which dilation transformation is shown in the graph. The graph represents a coordinate plane with a triangle \(\Delta XYZ\) and its image after a transformation.
### Description of the Graph:
1. **Original Triangle \(\Delta XYZ\)**:
- The triangle has its vertices labeled as \(X\), \(Y\), and \(Z\).
- Point \(X\) is located at coordinates \((2, 4)\).
- Point \(Y\) is at coordinates \((4, 4)\).
- Point \(Z\) is at coordinates \((2, 1)\).
2. **Transformed (Dilated) Triangle**:
- The image of \(\Delta XYZ\) after dilation has new vertices.
- Point \(X'\) is at coordinates \((6, 12)\).
- Point \(Y'\) is at coordinates \((12, 12)\).
- Point \(Z'\) is at coordinates \((6, 3)\).
### Multiple Choice Answers:
You need to determine which dilation corresponds to the transformation shown in the graph.
- **Option A:**
\[ D_{\frac{1}{4}}(\Delta XYZ) \]
- **Option B:**
\[ D_3(\Delta XYZ) \]
- **Option C:**
\[ D_{\frac{1}{3}}(\Delta XYZ) \]
- **Option D:**
\[ D_2(\Delta XYZ) \]
### Explanation:
To determine the correct dilation, we need to compare the coordinates of the original triangle's vertices with its image's vertices. The transformation factor (k) can be obtained using the formula for dilation \[ (kx, ky) \].
Reviewing the coordinates changes:
1. From \( (2, 4) \) to \( (6, 12) \), we observe that each coordinate is scaled up by a factor of 3.
2. From \( (4, 4) \) to \( (12, 12) \), we observe the same scaling factor of 3.
3. From \( (2, 1) \) to \( (6, 3) \), we again see the same scaling factor of 3.
The dilation that scales the figure by 3 times is \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb294f3d7-c37e-4d19-872e-5bb2e2baac27%2F728fbf79-3903-4965-90f3-dcfc68b784eb%2Fzrd9ixc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Identifying Dilation in a Graph**
In the given problem, we are tasked with identifying which dilation transformation is shown in the graph. The graph represents a coordinate plane with a triangle \(\Delta XYZ\) and its image after a transformation.
### Description of the Graph:
1. **Original Triangle \(\Delta XYZ\)**:
- The triangle has its vertices labeled as \(X\), \(Y\), and \(Z\).
- Point \(X\) is located at coordinates \((2, 4)\).
- Point \(Y\) is at coordinates \((4, 4)\).
- Point \(Z\) is at coordinates \((2, 1)\).
2. **Transformed (Dilated) Triangle**:
- The image of \(\Delta XYZ\) after dilation has new vertices.
- Point \(X'\) is at coordinates \((6, 12)\).
- Point \(Y'\) is at coordinates \((12, 12)\).
- Point \(Z'\) is at coordinates \((6, 3)\).
### Multiple Choice Answers:
You need to determine which dilation corresponds to the transformation shown in the graph.
- **Option A:**
\[ D_{\frac{1}{4}}(\Delta XYZ) \]
- **Option B:**
\[ D_3(\Delta XYZ) \]
- **Option C:**
\[ D_{\frac{1}{3}}(\Delta XYZ) \]
- **Option D:**
\[ D_2(\Delta XYZ) \]
### Explanation:
To determine the correct dilation, we need to compare the coordinates of the original triangle's vertices with its image's vertices. The transformation factor (k) can be obtained using the formula for dilation \[ (kx, ky) \].
Reviewing the coordinates changes:
1. From \( (2, 4) \) to \( (6, 12) \), we observe that each coordinate is scaled up by a factor of 3.
2. From \( (4, 4) \) to \( (12, 12) \), we observe the same scaling factor of 3.
3. From \( (2, 1) \) to \( (6, 3) \), we again see the same scaling factor of 3.
The dilation that scales the figure by 3 times is \(
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