Math worksheet with dilation problems involving triangles and quadrilaterals on a coordinate plane.

Problem Analysis:

The task involves performing dilations on geometric figures using specific scale factors and the origin as the center of dilation. Let's solve each part step by step.

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Problem 6: Dilate Triangle XYZ

#### Given:
- Scale factor: \( k = 1.5 \)
- Center of dilation: \( (0, 0) \)
- Coordinates of triangle \( XYZ \):
- \( X(-4, 3) \)
- \( Y(6, 0) \)
- \( Z(-2, -3) \)

#### Formula for Dilation:
The coordinates of a point \( (x, y) \) after dilation with scale factor \( k \) and center \( (0, 0) \) are:
\[
(x', y') = (k \cdot x, k \cdot y)
\]

#### Step 1: Calculate the coordinates of the dilated points.
1. Point \( X(-4, 3) \):
\[
X' = (1.5 \cdot (-4), 1.5 \cdot 3) = (-6, 4.5)
\]

2. Point \( Y(6, 0) \):
\[
Y' = (1.5 \cdot 6, 1.5 \cdot 0) = (9, 0)
\]

3. Point \( Z(-2, -3) \):
\[
Z' = (1.5 \cdot (-2), 1.5 \cdot (-3)) = (-3, -4.5)
\]

#### Final Answer for Problem 6:
\[
\boxed{
\begin{aligned}
X &: (-4, 3) & X' &: (-6, 4.5) \\
Y &: (6, 0) & Y' &: (9, 0) \\
Z &: (-2, -3) & Z' &: (-3, -4.5)
\end{aligned}
}
\]

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Problem 7: Dilate Quadrilateral MNOP

#### Given:
- Scale factor: \( k = \frac{1}{3} \)
- Center of dilation: \( (0, 0) \)
- Coordinates of quadrilateral \( MNOP \):
- \( M(0, 6) \)
- \( N(6, 6) \)
- \( O(5, 0) \)
- \( P(-3, 0) \)

#### Step 1: Calculate the coordinates of the dilated points.
1. Point \( M(0, 6) \):
\[
M' = \left( \frac{1}{3} \cdot 0, \frac{1}{3} \cdot 6 \right) = (0, 2)
\]

2. Point \( N(6, 6) \):
\[
N' = \left( \frac{1}{3} \cdot 6, \frac{1}{3} \cdot 6 \right) = (2, 2)
\]

3. Point \( O(5, 0) \):
\[
O' = \left( \frac{1}{3} \cdot 5, \frac{1}{3} \cdot 0 \right) = \left( \frac{5}{3}, 0 \right)
\]

4. Point \( P(-3, 0) \):
\[
P' = \left( \frac{1}{3} \cdot (-3), \frac{1}{3} \cdot 0 \right) = (-1, 0)
\]

#### Final Answer for Problem 7:
\[
\boxed{
\begin{aligned}
M &: (0, 6) & M' &: (0, 2) \\
N &: (6, 6) & N' &: (2, 2) \\
O &: (5, 0) & O' &: \left( \frac{5}{3}, 0 \right) \\
P &: (-3, 0) & P' &: (-1, 0)
\end{aligned}
}
\]

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Problem 8: Describe the Dilation of Quadrilateral MNOP

#### Given:
- The image shows the original quadrilateral \( MNOP \) and its dilated image \( M'N'O'P' \).
- The center of dilation is the origin \( (0, 0) \).

#### Step 1: Identify the scale factor.
From the graph, observe the relationship between the original points and their images:
- Original point \( M(0, 6) \) maps to \( M'(0, 2) \).
- The y-coordinate of \( M \) is scaled from 6 to 2, which is a factor of \( \frac{2}{6} = \frac{1}{3} \).

Similarly, check other points:
- \( N(6, 6) \) maps to \( N'(2, 2) \).
- \( O(5, 0) \) maps to \( O'\left( \frac{5}{3}, 0 \right) \).
- \( P(-3, 0) \) maps to \( P'(-1, 0) \).

In all cases, the coordinates are scaled by \( \frac{1}{3} \).

#### Step 2: Describe the dilation.
The quadrilateral \( MNOP \) is dilated by a scale factor of \( \frac{1}{3} \) with the origin \( (0, 0) \) as the center of dilation. This means every point on the quadrilateral is moved closer to the origin by a factor of \( \frac{1}{3} \).

#### Final Answer for Problem 8:
\[
\boxed{\text{The quadrilateral } MNOP \text{ is dilated by a scale factor of } \frac{1}{3} \text{ with the origin as the center of dilation.}}
\]

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Summary of Answers:

1. Problem 6:
\[
\boxed{
\begin{aligned}
X &: (-4, 3) & X' &: (-6, 4.5) \\
Y &: (6, 0) & Y' &: (9, 0) \\
Z &: (-2, -3) & Z' &: (-3, -4.5)
\end{aligned}
}
\]

2. Problem 7:
\[
\boxed{
\begin{aligned}
M &: (0, 6) & M' &: (0, 2) \\
N &: (6, 6) & N' &: (2, 2) \\
O &: (5, 0) & O' &: \left( \frac{5}{3}, 0 \right) \\
P &: (-3, 0) & P' &: (-1, 0)
\end{aligned}
}
\]

3. Problem 8:
\[
\boxed{\text{The quadrilateral } MNOP \text{ is dilated by a scale factor of } \frac{1}{3} \text{ with the origin as the center of dilation.}}
\]