Free Interactive & Printable Dilations Worksheet Collection - Free Printable

Let's solve each dilation problem step by step. Dilation is a transformation that changes the size of a figure while keeping its shape and orientation. The center of dilation is the fixed point from which all points are scaled, and the scale factor determines how much larger or smaller the image becomes.

We'll use the formula for dilation:

> If a point $ P $ has coordinates $ (x, y) $, and the center of dilation is $ C = (a, b) $ with scale factor $ k $, then the image point $ P' $ is:
>
> $$
> P' = (a + k(x - a),\ b + k(y - b))
> $$

This means:
- Find the vector from the center to the point: $ (x - a, y - b) $
- Multiply this vector by the scale factor $ k $
- Add it back to the center coordinates

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Problem 5: Dilate hexagon QRSTUV

- Center of dilation: Point V
- Scale factor: 2

#### Step 1: Identify coordinates of points from the graph

From the grid:
- Q: (-1, 0)
- R: (0, 1)
- S: (1, 1)
- T: (2, 0)
- U: (1, -1)
- V: (0, -1)

Center of dilation: V = (0, -1)

Now apply dilation with scale factor 2.

For each point:
$$
P' = V + 2 \times (P - V)
$$

- Q':
$$
Q = (-1, 0),\ V = (0, -1) \\
Q - V = (-1, 1) \\
2 \times (-1, 1) = (-2, 2) \\
Q' = (0, -1) + (-2, 2) = (-2, 1)
$$

- R':
$$
R = (0, 1),\ R - V = (0, 2) \\
2 \times (0, 2) = (0, 4) \\
R' = (0, -1) + (0, 4) = (0, 3)
$$

- S':
$$
S = (1, 1),\ S - V = (1, 2) \\
2 \times (1, 2) = (2, 4) \\
S' = (0, -1) + (2, 4) = (2, 3)
$$

- T':
$$
T = (2, 0),\ T - V = (2, 1) \\
2 \times (2, 1) = (4, 2) \\
T' = (0, -1) + (4, 2) = (4, 1)
$$

- U':
$$
U = (1, -1),\ U - V = (1, 0) \\
2 \times (1, 0) = (2, 0) \\
U' = (0, -1) + (2, 0) = (2, -1)
$$

- V':
Since V is the center, it stays the same:
$$
V' = (0, -1)
$$

✔ Answer for Problem 5:
- Q': (-2, 1)
- R': (0, 3)
- S': (2, 3)
- T': (4, 1)
- U': (2, -1)
- V': (0, -1)

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Problem 6: Dilate trapezoid WXYZ

- Center of dilation: Point Z
- Scale factor: $ \frac{1}{2} $

#### Coordinates from graph:
- W: (1, 2)
- X: (3, 2)
- Y: (5, 0)
- Z: (0, 0)

Center: Z = (0, 0)

Apply dilation with scale factor $ \frac{1}{2} $

$$
P' = Z + \frac{1}{2}(P - Z) = \frac{1}{2}P \quad \text{(since Z is origin)}
$$

- W':
$$
W = (1, 2) \Rightarrow W' = \left(\frac{1}{2}, 1\right)
$$

- X':
$$
X = (3, 2) \Rightarrow X' = \left(\frac{3}{2}, 1\right)
$$

- Y':
$$
Y = (5, 0) \Rightarrow Y' = \left(\frac{5}{2}, 0\right) = (2.5, 0)
$$

- Z':
Center stays same → Z' = (0, 0)

✔ Answer for Problem 6:
- W': (0.5, 1)
- X': (1.5, 1)
- Y': (2.5, 0)
- Z': (0, 0)

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Problem 7: Dilate rectangle EFGH

- Center of dilation: Point F
- Scale factor: $ \frac{3}{4} $

#### Coordinates:
- E: (-1, 3)
- F: (0, 3)
- G: (0, 1)
- H: (-1, 1)

Center: F = (0, 3)

Use:
$$
P' = F + \frac{3}{4}(P - F)
$$

- E':
$$
E = (-1, 3),\ E - F = (-1, 0) \\
\frac{3}{4}(-1, 0) = (-0.75, 0) \\
E' = (0, 3) + (-0.75, 0) = (-0.75, 3)
$$

- F':
Center → F' = (0, 3)

- G':
$$
G = (0, 1),\ G - F = (0, -2) \\
\frac{3}{4}(0, -2) = (0, -1.5) \\
G' = (0, 3) + (0, -1.5) = (0, 1.5)
$$

- H':
$$
H = (-1, 1),\ H - F = (-1, -2) \\
\frac{3}{4}(-1, -2) = (-0.75, -1.5) \\
H' = (0, 3) + (-0.75, -1.5) = (-0.75, 1.5)
$$

✔ Answer for Problem 7:
- E': (-0.75, 3)
- F': (0, 3)
- G': (0, 1.5)
- H': (-0.75, 1.5)

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Problem 8: Dilate hexagon LMNOPQ

- Center of dilation: Point M
- Scale factor: 3

#### Coordinates:
- L: (6, 3)
- M: (6, 1)
- N: (5, 1)
- O: (5, 0)
- P: (4, 0)
- Q: (4, 1)

Center: M = (6, 1)

Use:
$$
P' = M + 3(P - M)
$$

- L':
$$
L = (6, 3),\ L - M = (0, 2) \\
3(0, 2) = (0, 6) \\
L' = (6, 1) + (0, 6) = (6, 7)
$$

- M':
Center → M' = (6, 1)

- N':
$$
N = (5, 1),\ N - M = (-1, 0) \\
3(-1, 0) = (-3, 0) \\
N' = (6, 1) + (-3, 0) = (3, 1)
$$

- O':
$$
O = (5, 0),\ O - M = (-1, -1) \\
3(-1, -1) = (-3, -3) \\
O' = (6, 1) + (-3, -3) = (3, -2)
$$

- P':
$$
P = (4, 0),\ P - M = (-2, -1) \\
3(-2, -1) = (-6, -3) \\
P' = (6, 1) + (-6, -3) = (0, -2)
$$

- Q':
$$
Q = (4, 1),\ Q - M = (-2, 0) \\
3(-2, 0) = (-6, 0) \\
Q' = (6, 1) + (-6, 0) = (0, 1)
$$

✔ Answer for Problem 8:
- L': (6, 7)
- M': (6, 1)
- N': (3, 1)
- O': (3, -2)
- P': (0, -2)
- Q': (0, 1)

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✔ Final Answers Summary:

#### Problem 5:
- Q': (-2, 1)
- R': (0, 3)
- S': (2, 3)
- T': (4, 1)
- U': (2, -1)
- V': (0, -1)

#### Problem 6:
- W': (0.5, 1)
- X': (1.5, 1)
- Y': (2.5, 0)
- Z': (0, 0)

#### Problem 7:
- E': (-0.75, 3)
- F': (0, 3)
- G': (0, 1.5)
- H': (-0.75, 1.5)

#### Problem 8:
- L': (6, 7)
- M': (6, 1)
- N': (3, 1)
- O': (3, -2)
- P': (0, -2)
- Q': (0, 1)

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