Dilations: Step-by-Step Examples for Visual Learners

Understanding Dilations: A Visual Guide 📐

Dilation is a transformation that changes the size of a figure. It either enlarges or reduces the figure. The amount of enlargement or reduction is determined by the scale factor. Let's break it down with examples.

Key Concepts:

• Center of Dilation: The fixed point from which the figure is enlarged or reduced.
• Scale Factor (k): The ratio of the new image size to the original image size.
• If k > 1, the image is an enlargement.
• If 0 < k < 1, the image is a reduction.
• If k = 1, the image remains unchanged.

Step-by-Step Examples 🚀

Example 1: Enlargement

Consider a triangle with vertices A(1, 1), B(2, 1), and C(1, 2). We want to dilate this triangle by a scale factor of 2, with the center of dilation at the origin (0, 0).

1. Multiply each coordinate by the scale factor:
• A'(2 * 1, 2 * 1) = A'(2, 2)
• B'(2 * 2, 2 * 1) = B'(4, 2)
• C'(2 * 1, 2 * 2) = C'(2, 4)
2. Plot the new points: A'(2, 2), B'(4, 2), and C'(2, 4).
3. Connect the points to form the dilated triangle. The new triangle is twice the size of the original.

# Python code to perform dilation
def dilate(point, scale_factor, center=(0, 0)):
    x, y = point
    center_x, center_y = center
    
    # Translate point to origin
    translated_x = x - center_x
    translated_y = y - center_y
    
    # Scale the translated point
    scaled_x = translated_x * scale_factor
    scaled_y = translated_y * scale_factor
    
    # Translate back to original position
    final_x = scaled_x + center_x
    final_y = scaled_y + center_y
    
    return (final_x, final_y)

# Example usage
point = (1, 1)
scale_factor = 2
dilated_point = dilate(point, scale_factor)
print(dilated_point) # Output: (2.0, 2.0)

Example 2: Reduction

Let's take a square with vertices P(4, 4), Q(8, 4), R(8, 8), and S(4, 8). We want to reduce this square by a scale factor of 0.5, with the center of dilation at the origin (0, 0).

1. Multiply each coordinate by the scale factor:
• P'(0.5 * 4, 0.5 * 4) = P'(2, 2)
• Q'(0.5 * 8, 0.5 * 4) = Q'(4, 2)
• R'(0.5 * 8, 0.5 * 8) = R'(4, 4)
• S'(0.5 * 4, 0.5 * 8) = S'(2, 4)
2. Plot the new points: P'(2, 2), Q'(4, 2), R'(4, 4), and S'(2, 4).
3. Connect the points to form the dilated square. The new square is half the size of the original.

Example 3: Dilation with a different center

Consider a point A(2, 3) and we want to dilate it by a scale factor of 3 with the center of dilation at (1, 1).

1. Translate the point so the center of dilation is at the origin: A' = (2-1, 3-1) = (1, 2)
2. Multiply by the scale factor: A'' = (3*1, 3*2) = (3, 6)
3. Translate back: A''' = (3+1, 6+1) = (4, 7)

Important Considerations 🤔

• If the scale factor is negative, the image is also reflected across the center of dilation.
• Dilation preserves the shape of the figure, but not the size.
• Angles remain unchanged after dilation.

Practice Makes Perfect ✍️

Try dilating different shapes with various scale factors and centers of dilation. Visualizing the transformation will help you understand the concept better. Good luck!