Lesson 8-4: Dilations

Property

A dilation is a transformation that changes the size of a figure by a scale factor $k$ with respect to a fixed point called the center of dilation.

All points on the original figure are mapped to corresponding points on the image such that the distance from the center to each image point is $k$ times the distance from the center to the corresponding original point.

Property

For a dilation with scale factor $k > 0$, the value of $k$ determines how the size of the image compares to the preimage:

• If $k > 1$, the dilation is an enlargement.
• If $0 < k < 1$, the dilation is a reduction.
• If $k = 1$, the dilation is a congruence transformation (the image and preimage are congruent).

Examples

Property

When performing a dilation on a coordinate grid where the Center of Dilation is the origin (0,0), the rule is the easiest of all transformations: simply multiply both the x and y coordinates of every vertex by the scale factor k.

Rule: (x, y) → (kx, ky)

Examples

• Enlargement on Grid: Dilate point A(-3, 5) with a scale factor of k = 2.
  • New x: -3 * 2 = -6
  • New y: 5 * 2 = 10
  • Image: A'(-6, 10)
• Reduction on Grid: Triangle JKL has a vertex at J(4, -8). Dilate it by k = 1/2.
  • New x: 4 * (1/2) = 2
  • New y: -8 * (1/2) = -4
  • Image: J'(2, -4)

Property

To verify a dilation with center at origin and scale factor $k$:

1. Check that each image coordinate equals $k$ times the original coordinate: $(x', y') = (kx, ky)$
2. Confirm that the ratio of distances from origin is constant: $\frac{\text{distance to image point}}{\text{distance to original point}} = |k|$
3. Verify that original and image points lie on the same ray from the origin

Examples