Understanding Scale Factor in Dilation for Beginners

To find the scale factor of the dilation from figure ABCD to figure A'B'C'D, follow these simple steps:

1. Identify a Point: Choose a corresponding point from figure ABCD and its dilated counterpart. For example, let point A be at coordinates (x, y) and point A' at (x', y').
2. Calculate the Distance: Since the center of dilation is at the origin (0, 0), calculate the distance from the origin to both points:
• Distance to point A: d_A = √(x² + y²)
• Distance to point A': d_A' = √(x'² + y'²)
3. Determine the Scale Factor: The scale factor (k) can be calculated using the formula:
k = d_A' / d_A
4. Conclusion: The value of k will tell you how much the figure was enlarged or reduced. If k > 1, the figure is enlarged; if k < 1, the figure is reduced.

For example, if Point A is at (2, 3) and Point A' is at (4, 6), then:

• Distance to A: d_A = √(2² + 3²) = √(4 + 9) = √13
• Distance to A': d_A' = √(4² + 6²) = √(16 + 36) = √52
• Scale factor: k = √52 / √13 = 2

This means the figure is dilated by a scale factor of 2.