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Determine scale factors of dilations

A scale factor is the constant ratio of each image length to its corresponding preimage length, determined from corresponding side lengths, coordinates, or distances from the center of dilation. It explains enlargements when the factor is greater than 1 and reductions when it lies between 0 and 1; for a dilation centered at the origin, coordinates are multiplied by the factor, preserving angle measures and proportional relationships. The focus is on positive factors, not signed factors or more advanced matrix and higher-dimensional treatments.

Detailed Explanation: Determine scale factors of dilations

A scale factor tells how much every length changes in a dilation.

Use the formula

scale factor=image lengthpreimage length.\text{scale factor}=\frac{\text{image length}}{\text{preimage length}}.

The preimage is the original figure, and the image is the dilated figure.

  • If the scale factor is greater than 11, the figure is enlarged.
  • If the scale factor is between 00 and 11, the figure is reduced.

Example: A side of a triangle is 66 centimeters in the preimage and 99 centimeters in the image. Find the scale factor.

  1. Identify the corresponding lengths:

    • Preimage: 66 cm
    • Image: 99 cm

  2. Divide the image length by the preimage length:

k=96k=\frac{9}{6}
  1. Simplify:
k=32=1.5k=\frac{3}{2}=1.5

The scale factor is

1.5\boxed{1.5}

This means every side length in the image is 1.51.5 times the corresponding side length in the preimage, so the dilation is an enlargement. For example, a preimage side of 88 cm would become

1.5(8)=12 cm.1.5(8)=12\text{ cm}.

Learn by doing: Determine scale factors of dilations

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    Dilations - Grid Images & Side Measurements to Scale Factor

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