By Allen Ma, Amber Kuang

In geometry, a dilation is a transformation that changes only the size of a geometric shape, while leaving its shape and orientation unchanged. In the following practice questions, you’re asked to calculate the constant of dilation, and then find the dilated image of given coordinates.

The center of dilation is the origin.

Practice questions

  1. If Point

    image0.png

    is the image of Point A (3, 2) under a dilation with respect to the origin, what is the constant of dilation?

  2. A dilation maps J (6, 3) to

    image1.png

    What would the image of K (–2, 10) be under the same dilation?

Answers and explanations

  1. 3

    A dilation changes the distance between points by multiplying the x and y coordinates by the scale factor. The x coordinate for A is 3, and the x coordinate for

    image2.png

    is 9. If you call the scalar factor k, then

    image3.png

    You can also determine the scale factor by looking at the y coordinates. The y coordinate for A is 2, and the y coordinate for

    image4.png

    is 6, so

    image5.png

  2. (–1, 5)

    A dilation changes the distance between points by multiplying the x and y coordinates by the scale factor. The x coordinate for J is 6, and the x coordinate for

    image6.png

    is 3. If you call the scalar factor k, then

    image7.png

    Multiply the coordinates of K by the scale factor of 1/2:

    image8.png

    The image is therefore (–1, 5).