Free Interactive & Printable Dilations Worksheet Collection - Free Printable

This worksheet focuses on dilations on the coordinate plane. A dilation is a transformation that changes the size of a figure by a scale factor relative to a center of dilation (in this case, the origin $(0, 0)$). The rule for dilating a point $(x, y)$ with a scale factor $k$ centered at the origin is:

$$
(x', y') = (k \cdot x, k \cdot y)
$$

Let’s go through each problem and verify or explain the solutions.

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1. Dilate parallelogram ABCD

- Center of dilation: $(0, 0)$
- Scale factor: $\frac{1}{3}$

We apply the rule: multiply each coordinate by $\frac{1}{3}$.

From the graph:
- $A = (-9, 6)$ → $A' = \left(\frac{1}{3} \cdot -9, \frac{1}{3} \cdot 6\right) = (-3, 2)$ ✔
- $B = (-3, 6)$ → $B' = \left(\frac{1}{3} \cdot -3, \frac{1}{3} \cdot 6\right) = (-1, 2)$ ✔
- $C = (0, 3)$ → $C' = \left(\frac{1}{3} \cdot 0, \frac{1}{3} \cdot 3\right) = (0, 1)$ ✔
- $D = (-6, 3)$ → $D' = \left(\frac{1}{3} \cdot -6, \frac{1}{3} \cdot 3\right) = (-2, 1)$ ✔

✔ All coordinates are correct.

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2. Dilate kite EFGH

- Center of dilation: $(0, 0)$
- Scale factor: $2$

Multiply each coordinate by 2.

From the graph:
- $E = (-3, -2)$ → $E' = (-6, -4)$ ✔
- $F = (0, 4)$ → $F' = (0, 8)$ ✔
- $G = (3, -2)$ → $G' = (6, -4)$ ✔
- $H = (0, -5)$ → $H' = (0, -10)$ ✔

✔ All coordinates are correct.

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3. Dilate triangle IJK

- Center of dilation: $(0, 0)$
- Scale factor: $\frac{2}{3}$

Multiply each coordinate by $\frac{2}{3}$.

From the graph:
- $I = (-9, -3)$ → $I' = \left(\frac{2}{3} \cdot -9, \frac{2}{3} \cdot -3\right) = (-6, -2)$ ✔
- $J = (6, -3)$ → $J' = \left(\frac{2}{3} \cdot 6, \frac{2}{3} \cdot -3\right) = (4, -2)$ ✔
- $K = (-9, -9)$ → $K' = \left(\frac{2}{3} \cdot -9, \frac{2}{3} \cdot -9\right) = (-6, -6)$ ✔

✔ All coordinates are correct.

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4. Dilate triangle LMN

- Center of dilation: $(0, 0)$
- Scale factor: $\frac{3}{2}$

Multiply each coordinate by $\frac{3}{2}$.

From the graph:
- $L = (-6, 0)$ → $L' = \left(\frac{3}{2} \cdot -6, \frac{3}{2} \cdot 0\right) = (-9, 0)$ ✔
- $M = (0, -6)$ → $M' = \left(\frac{3}{2} \cdot 0, \frac{3}{2} \cdot -6\right) = (0, -9)$ ✔
- $N = (-4, -4)$ → $N' = \left(\frac{3}{2} \cdot -4, \frac{3}{2} \cdot -4\right) = (-6, -6)$ ✔

✔ All coordinates are correct.

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✔ Summary:

All the given answers on the worksheet are correct. Each image point was found using the dilation formula:
$$
(x', y') = (k \cdot x, k \cdot y)
$$
where $k$ is the scale factor and the center is at the origin.

The red points shown in the graphs represent the dilated images, and their coordinates match the calculated values.

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🔍 Key Concepts:

- Dilation changes the size but not the shape.
- If $|k| > 1$: enlargement
- If $0 < |k| < 1$: reduction
- If $k < 0$: reflection over the origin + scaling

In this worksheet:
- Problem 1: Reduction ($\frac{1}{3}$)
- Problem 2: Enlargement ($2$)
- Problem 3: Reduction ($\frac{2}{3}$)
- Problem 4: Enlargement ($\frac{3}{2}$)

All transformations are centered at the origin, so the calculations are straightforward multiplications.

✔ Final Answer: All provided coordinates are correct.