I need help **Exercise: Dilations in the Coordinate Plane**---**Problem 3:**Given that line segment $ \overline{AB} $ has coordinates $ A(-3, 9) $ and $ B(6, -12) $, determine the coordinates of $ A'B' $ after a dilation with a scale factor of $\frac{2}{3}$ centered at the origin.---**Diagram Explanation:**The provided graph on the right side of the document consists of a standard Cartesian coordinate plane. The x-axis and y-axis intersect at the origin $(0, 0)$. The axes extend from -9 to 9 with grid lines clearly marked, providing a visual aid for plotting points and performing transformations such as dilations.---When performing the dilation:- Multiply each coordinate of the points by the scale factor $\frac{2}{3}$.- New coordinates for $ A' $ would be $\left(\frac{2}{3} \times -3, \frac{2}{3} \times 9\right)$.- New coordinates for $ B' $ would be $\left(\frac{2}{3} \times 6, \frac{2}{3} \times -12\right)$.Carefully plot these new points on the graph to visualize the dilation effect. # Problem Statement**Question 4:**Line segment $ PQ $ has coordinates $ P(3, -6) $ and $ Q(3, 9) $. Find the coordinates of $ P'Q' $ after a dilation with a scale factor of $\frac{1}{5}$ centered at $ (3, -1) $.# Graph ExplanationThe image includes a coordinate grid that displays both negative and positive values on the x and y axes. Each axis is marked with units ranging from -9 to 9. The grid can be used to plot the coordinates of points and visualize transformations such as the dilation described in the problem statement.

Given: 1. AB¯ has coordinates A-3,9 and B6,-12. 2. PQ¯ has coordinates P3,-6 and Q3,9. To Find: 1.…

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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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**Exercise: Dilations in the Coordinate Plane**

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**Problem 3:**

Given that line segment $ \overline{AB} $ has coordinates $ A(-3, 9) $ and $ B(6, -12) $, determine the coordinates of $ A'B' $ after a dilation with a scale factor of $\frac{2}{3}$ centered at the origin.

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**Diagram Explanation:**

The provided graph on the right side of the document consists of a standard Cartesian coordinate plane. The x-axis and y-axis intersect at the origin $(0, 0)$. The axes extend from -9 to 9 with grid lines clearly marked, providing a visual aid for plotting points and performing transformations such as dilations.

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When performing the dilation:

- Multiply each coordinate of the points by the scale factor $\frac{2}{3}$.
- New coordinates for $ A' $ would be $\left(\frac{2}{3} \times -3, \frac{2}{3} \times 9\right)$.
- New coordinates for $ B' $ would be $\left(\frac{2}{3} \times 6, \frac{2}{3} \times -12\right)$.

Carefully plot these new points on the graph to visualize the dilation effect.

# Problem Statement

**Question 4:**

Line segment $ PQ $ has coordinates $ P(3, -6) $ and $ Q(3, 9) $. Find the coordinates of $ P'Q' $ after a dilation with a scale factor of $\frac{1}{5}$ centered at $ (3, -1) $.

# Graph Explanation

The image includes a coordinate grid that displays both negative and positive values on the x and y axes. Each axis is marked with units ranging from -9 to 9. The grid can be used to plot the coordinates of points and visualize transformations such as the dilation described in the problem statement.

Expert Solution

Step 1

Given:

$1 . \bar{A B} has coordinates A (- 3, 9) and B (6, - 12) . 2 . \bar{P Q} has coordinates P (3, - 6) and Q (3, 9) .$

To Find:

$1 . Find the coordiante \bar{A' B'} after a dilation with a scale factor of \frac{2}{3} centered at the origin. 2 . Find the coordiante \bar{P' Q'} after a dilation with a scale factor of \frac{1}{5} centered at the (3, - 1) .$

Concept:

$1 . Dilation with center at Origin: D_{O, k} (x, y) = (k x, k y), where O = center of dilation at (0, 0) k = scale factor 2 . Dilation with center not at Origin: D_{O, k} (x, y) = (k (x - a) + a, k (y - b) + b), where O = center of dilation at (a, b) k = scale factor$

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