Grade 8 Geometry: Dilations - How to Scale Shapes

📐 Understanding Dilations in Geometry

Dilation is a transformation that changes the size of a figure. It either enlarges (stretches) or reduces (shrinks) the figure. Two key components define a dilation:

• Center of Dilation: A fixed point around which the figure is enlarged or reduced.
• Scale Factor: A number that determines the amount of enlargement or reduction.

📏 How to Scale Shapes

To dilate a shape, you need to determine the coordinates of the original shape (pre-image), the center of dilation, and the scale factor.

1. Identify the Coordinates: Note the coordinates of each vertex of the pre-image.
2. Determine the Center of Dilation: This is the fixed point. Often, it's the origin (0,0).
3. Apply the Scale Factor: Multiply the coordinates of each vertex by the scale factor.

✍️ Formula for Dilation

If the center of dilation is at the origin (0,0), the formula to find the new coordinates (x', y') after dilation is:

$(x', y') = (kx, ky)$

Where:

• $(x, y)$ are the original coordinates.
• $k$ is the scale factor.

🔢 Examples

Example 1: Dilation with Center at Origin

Suppose we have a triangle with vertices A(1, 2), B(3, 4), and C(5, 2). We want to dilate it by a scale factor of 2 with the center of dilation at the origin.

New coordinates:

• A'(2*1, 2*2) = A'(2, 4)
• B'(2*3, 2*4) = B'(6, 8)
• C'(2*5, 2*2) = C'(10, 4)

Example 2: Dilation with Center at Origin (Reduction)

Suppose we have a square with vertices P(4, 4), Q(4, 8), R(8, 8), and S(8, 4). We want to dilate it by a scale factor of 0.5 (a reduction) with the center of dilation at the origin.

New coordinates:

• P'(0.5*4, 0.5*4) = P'(2, 2)
• Q'(0.5*4, 0.5*8) = Q'(2, 4)
• R'(0.5*8, 0.5*8) = R'(4, 4)
• S'(0.5*8, 0.5*4) = S'(4, 2)

💻 Code Example (Python)

Here's a simple Python function to perform dilation:

def dilate(x, y, k):
    new_x = k * x
    new_y = k * y
    return new_x, new_y

# Example usage
x, y = 1, 2
k = 2
new_x, new_y = dilate(x, y, k)
print(f"Original coordinates: ({x}, {y})")
print(f"New coordinates after dilation: ({new_x}, {new_y})")

💡 Key Takeaways

• Dilation changes the size of a shape.
• The scale factor determines the extent of enlargement or reduction.
• The center of dilation is the fixed point for the transformation.