16.4 - Dilation

Concept summary and connections

• Dilation is scaling - you change the size of a diagram, but keep all of the angles and length ratios constant.
• Dilation has a center and a scale factor. Every part of the diagram is changed:
  • A vector connecting the center to any part of the diagram will be parallel to the vector from the center to the corresponding part of the image
  • The vector connecting the center to the image will be scaled in length according to the scale factor (scale factor 2 means twice as long as the original, for example)
  • A negative scale factor means that the vector is pointing antiparallel to the original (opposite direction along the same line).

Media resources

• Youtube search for "Dilation is scaling - you change the size of a diagram, but keep all of the angles and length ratios constant."
• Youtube search for "Dilation has a center and a scale factor. Every part of the diagram is changed so that a ray drawn from the center to any point on it will be parallel to the original, but its length is scaled by the dilation factor."

Guided practice

• How does the area of a square change if it is dilated by a factor of 2? How about by 1/2?
• What happens when you perform a dilation with a scale factor of -1?
• Triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ is the image of a dilation of triangle $ABC$ about the centroid of triangle $ABC$. The dilation has a positive scale factor.
  • (a) Suppose $\mathrm{△}{A}^{\prime }{B}^{\prime }{C}^{\prime }$ is entirely outside $\mathrm{△}ABC$. Is the scale factor greater than 1 or less than 1?
  • (b) Suppose $\mathrm{△}{A}^{\prime }{B}^{\prime }{C}^{\prime }$ is entirely inside $\mathrm{△}ABC$. Is the scale factor greater than 1 or less than 1?
  • (c) What happens if the scale factor equals 1?
• Square $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ is the image of a dilation of square $ABCD$ about $A$ with scale factor 3. Suppose $AB=2$.
  • (a) Find $A{A}^{\prime }$, $A{B}^{\prime }$, and $B{B}^{\prime }$.
  • (b) Find $B{C}^{\prime }$ and $C{D}^{\prime }$.
• Let $A^{\prime }$ and $B^{\prime }$ be the images of points $A$ and $B$, respectively, under a dilation with center $O$ and positive scale factor $k$.
  • (a) Show that $\mathrm{△}AOB\sim \mathrm{△}{A}^{\prime }O{B}^{\prime }$.
  • (b) Show that $\overline{AB}\parallel \overline{A^{\prime }{B}^{\prime }}$ and $A^{\prime }{B}^{\prime }/AB=k$.

Homework

• #hw (geom-gr10) 16.4 Dilation (p. 418) (gr10) 2025-04-02