Learn on PengiReveal Math, AcceleratedUnit 6: Congruence and Similarity

Lesson 6-5: Explore Dilations

In this Grade 7 lesson from Reveal Math, Accelerated, students explore dilations as a type of transformation that changes the size of a figure using a scale factor, distinguishing them from rigid motion transformations. Students practice applying dilations to figures on and off the coordinate plane, including enlargements and reductions, by multiplying side lengths or coordinates by a given scale factor with the origin as the center of dilation. The lesson covers identifying whether a dilation is an enlargement or reduction and determining unknown scale factors from pre-image and image dimensions.

Section 1

Defining Dilations

Property

A dilation is a transformation that changes the size of a figure by a scale factor kk 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 kk times the distance from the center to the corresponding original point.

Section 2

Defining the Scale Factor (k): Enlargements and Reductions

Property

For a dilation with scale factor kk:

  • If k>1k > 1, the dilation is an enlargement (image is larger than original)
  • If 0<k<10 < k < 1, the dilation is a reduction (image is smaller than original)
  • If k=1k = 1, the image is congruent to the original figure

Examples

Section 3

Coordinate Rules for Dilations (Centered at Origin)

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)

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Dilations

Property

A dilation is a transformation that changes the size of a figure by a scale factor kk 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 kk times the distance from the center to the corresponding original point.

Section 2

Defining the Scale Factor (k): Enlargements and Reductions

Property

For a dilation with scale factor kk:

  • If k>1k > 1, the dilation is an enlargement (image is larger than original)
  • If 0<k<10 < k < 1, the dilation is a reduction (image is smaller than original)
  • If k=1k = 1, the image is congruent to the original figure

Examples

Section 3

Coordinate Rules for Dilations (Centered at Origin)

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)