Worksheet with three dilation problems on coordinate planes.

Let's solve each problem step by step.

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Problem 6: Dilate Triangle XYZ with Scale Factor 1.5, Center at (0,0)

We are given triangle $ XYZ $ and need to dilate it using a scale factor of 1.5 with the origin (0,0) as the center of dilation.

#### Step 1: Find Coordinates of Original Points
From the graph:

- $ X = (-4, 3) $
- $ Y = (6, 0) $
- $ Z = (-2, -2) $

#### Step 2: Apply Dilation Formula
When dilating from the origin with scale factor $ k $, the rule is:
$$
(x, y) \rightarrow (kx, ky)
$$
Here, $ k = 1.5 = \frac{3}{2} $

So,

- $ X' = (1.5 \times -4, 1.5 \times 3) = (-6, 4.5) $
- $ Y' = (1.5 \times 6, 1.5 \times 0) = (9, 0) $
- $ Z' = (1.5 \times -2, 1.5 \times -2) = (-3, -3) $

#### ✔ Answers for Problem 6:
- $ X: (-4, 3) $ → $ X': (-6, 4.5) $
- $ Y: (6, 0) $ → $ Y': (9, 0) $
- $ Z: (-2, -2) $ → $ Z': (-3, -3) $

> You would plot these new points and connect them to form the dilated triangle $ X'Y'Z' $.

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Problem 7: Dilate Quadrilateral MNOP with Scale Factor 1/3, Center at Origin

Given quadrilateral $ MNOP $, scale factor $ \frac{1}{3} $, center at (0,0)

#### Step 1: Find Original Coordinates
From the graph:

- $ M = (3, 6) $
- $ N = (6, 0) $
- $ O = (3, -4) $
- $ P = (-3, 0) $

#### Step 2: Apply Dilation Rule
$ (x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right) $

So,

- $ M' = \left(\frac{1}{3} \times 3, \frac{1}{3} \times 6\right) = (1, 2) $
- $ N' = \left(\frac{1}{3} \times 6, \frac{1}{3} \times 0\right) = (2, 0) $
- $ O' = \left(\frac{1}{3} \times 3, \frac{1}{3} \times -4\right) = (1, -\frac{4}{3}) $
- $ P' = \left(\frac{1}{3} \times -3, \frac{1}{3} \times 0\right) = (-1, 0) $

#### ✔ Answers for Problem 7:
- $ M: (3, 6) $ → $ M': (1, 2) $
- $ N: (6, 0) $ → $ N': (2, 0) $
- $ O: (3, -4) $ → $ O': (1, -\frac{4}{3}) $
- $ P: (-3, 0) $ → $ P': (-1, 0) $

> Plot these points and connect them to get the smaller quadrilateral $ M'N'O'P' $.

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Problem 8: Describe the Dilation of Quadrilateral MNOP Using the Origin as Center

We are shown two quadrilaterals: original $ MNOP $ and image $ M'N'O'P' $. We are to describe the dilation.

#### Step 1: Identify Coordinates
From the graph:

Original:
- $ M = (3, 6) $
- $ N = (6, 0) $
- $ O = (3, -4) $
- $ P = (-3, 0) $

Image:
- $ M' = (1, 2) $
- $ N' = (2, 0) $
- $ O' = (1, -\frac{4}{3}) $
- $ P' = (-1, 0) $

#### Step 2: Check Scale Factor
Compare corresponding points:

- $ M \to M' $: $ (3,6) \to (1,2) $ → $ \frac{1}{3} $ of original
- $ N \to N' $: $ (6,0) \to (2,0) $ → $ \frac{1}{3} $
- $ O \to O' $: $ (3,-4) \to (1, -\frac{4}{3}) $ → $ \frac{1}{3} $
- $ P \to P' $: $ (-3,0) \to (-1,0) $ → $ \frac{1}{3} $

✔ All coordinates are multiplied by $ \frac{1}{3} $

#### ✔ Description:
The quadrilateral $ MNOP $ is dilated with a scale factor of $ \frac{1}{3} $ centered at the origin (0,0). This results in a smaller quadrilateral $ M'N'O'P' $ that is similar to the original, with all side lengths reduced to one-third of the original, and all vertices located along the same rays from the origin.

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✔ Final Answers Summary:

#### Problem 6:
- $ X: (-4, 3) $ → $ X': (-6, 4.5) $
- $ Y: (6, 0) $ → $ Y': (9, 0) $
- $ Z: (-2, -2) $ → $ Z': (-3, -3) $

#### Problem 7:
- $ M: (3, 6) $ → $ M': (1, 2) $
- $ N: (6, 0) $ → $ N': (2, 0) $
- $ O: (3, -4) $ → $ O': (1, -\frac{4}{3}) $
- $ P: (-3, 0) $ → $ P': (-1, 0) $

#### Problem 8:
> The quadrilateral $ MNOP $ is dilated by a scale factor of $ \frac{1}{3} $ with the origin as the center of dilation. The image $ M'N'O'P' $ is a smaller, similar figure, where each vertex is located $ \frac{1}{3} $ of the distance from the origin compared to the original.

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Let me know if you'd like a visual sketch or explanation on how to plot these!