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(-

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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.

Mathematics For Machine Technology

8th Edition

ISBN:9781337798310

Author:Peterson, John.

Publisher:Peterson, John.

Chapter87: An Introduction To G- And M-codes For Cnc Programming

Section: Chapter Questions

Problem 24A

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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(-

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