Given U(1,3), V(-4,-4), and W(-3,6) on the coordinate plane, perform a dilation of A UVW from center 0(0,0) 3 with a scale factor of-. Determine the coordinates of images of points U, V, and W, and describe how the coordinates of the image points are related to the coordinates of the pre-image points.
Given U(1,3), V(-4,-4), and W(-3,6) on the coordinate plane, perform a dilation of A UVW from center 0(0,0) 3 with a scale factor of-. Determine the coordinates of images of points U, V, and W, and describe how the coordinates of the image points are related to the coordinates of the pre-image points.
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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This is a geometry question.
![**Problem Statement: Dilation of a Triangle on the Coordinate Plane**
Given vertices of triangle \(UVW\) as follows:
- \( U(1, 3) \)
- \( V(-4, -4) \)
- \( W(-3, 6) \)
On the coordinate plane, perform a dilation of \( \Delta UVW \) with the center of dilation at the origin \( O(0,0) \).
**Task:**
Determine the coordinates of the image points \( U', V', \) and \( W' \) after the dilation, given a scale factor of \( \frac{3}{2} \). Describe how the coordinates of the image points are related to the coordinates of the pre-image points.
**Key Concepts:**
- **Dilation:** This is a transformation that produces an image that is the same shape as the original, but is a different size. The scale factor determines how much larger or smaller the image will be.
- **Scale Factor:** A number which scales, or multiplies, some quantity.
**Steps to Solve:**
1. **Identify the Original Coordinates:**
- \( U(1, 3) \)
- \( V(-4, -4) \)
- \( W(-3, 6) \)
2. **Determine the Scale Factor:** The scale factor is \( \frac{3}{2} \).
3. **Apply the Scale Factor to Each Coordinate:**
For a point \( (x, y) \), the new coordinates \( (x', y') \) after dilation by a scale factor \( k \) are given by:
\[
(x', y') = (kx, ky)
\]
- For \( U(1, 3) \):
\[
U' = \left( \frac{3}{2} \times 1, \frac{3}{2} \times 3 \right) = \left( \frac{3}{2}, \frac{9}{2} \right)
\]
- For \( V(-4, -4) \):
\[
V' = \left( \frac{3}{2} \times -4, \frac{3}{2} \times -4 \right) = \left( -6, -6 \right)
\]
- For \( W(-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bd2ac81-fb2a-4f7d-9c25-a067f0858fd4%2Ffd9b436f-5bec-4155-b2be-9f782cfd638c%2Fqpg91x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Dilation of a Triangle on the Coordinate Plane**
Given vertices of triangle \(UVW\) as follows:
- \( U(1, 3) \)
- \( V(-4, -4) \)
- \( W(-3, 6) \)
On the coordinate plane, perform a dilation of \( \Delta UVW \) with the center of dilation at the origin \( O(0,0) \).
**Task:**
Determine the coordinates of the image points \( U', V', \) and \( W' \) after the dilation, given a scale factor of \( \frac{3}{2} \). Describe how the coordinates of the image points are related to the coordinates of the pre-image points.
**Key Concepts:**
- **Dilation:** This is a transformation that produces an image that is the same shape as the original, but is a different size. The scale factor determines how much larger or smaller the image will be.
- **Scale Factor:** A number which scales, or multiplies, some quantity.
**Steps to Solve:**
1. **Identify the Original Coordinates:**
- \( U(1, 3) \)
- \( V(-4, -4) \)
- \( W(-3, 6) \)
2. **Determine the Scale Factor:** The scale factor is \( \frac{3}{2} \).
3. **Apply the Scale Factor to Each Coordinate:**
For a point \( (x, y) \), the new coordinates \( (x', y') \) after dilation by a scale factor \( k \) are given by:
\[
(x', y') = (kx, ky)
\]
- For \( U(1, 3) \):
\[
U' = \left( \frac{3}{2} \times 1, \frac{3}{2} \times 3 \right) = \left( \frac{3}{2}, \frac{9}{2} \right)
\]
- For \( V(-4, -4) \):
\[
V' = \left( \frac{3}{2} \times -4, \frac{3}{2} \times -4 \right) = \left( -6, -6 \right)
\]
- For \( W(-
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