Dilations on a Square Grid

Student Summary

Square grids can be useful for showing dilations, especially when the center of dilation and the point(s) being dilated lie at grid points. Rather than using a ruler to measure the distance between the points, we can count grid units.

For example, the dilation of point QQ with center of dilation PP and scale factor 32\frac32 will be 6 grid squares to the left and 3 grid squares down from PP, since QQ is 4 grid squares to the left and 2 grid squares down from PP . The dilated image is marked as Q’Q’.

Points P and Q and image point Q prime on a square grid. Let the lower left corner be (0 comma 0). Then the points are P(7 comma 5), Q(3 comma 3) and Q prime(1 comma 2).

Sometimes the square grid comes with coordinates, giving us a convenient way to name points. Sometimes the coordinates of the image can be found just using arithmetic, without having to measure.

For example, to perform a dilation with center of dilation at (0,0)(0,0) and scale factor 2 on the triangle with coordinates (-1,-2)(\text-1, \text-2), (3,1)(3,1), and (2,-1)(2, \text-1), we can just double the coordinates to get (-2,-4)(\text-2, \text-4), (6,2)(6,2), and (4,-2)(4, \text-2).

Dilation on a coordinate plane, origin O.
Two triangles on a coordinate plane, origin O. Horizontal axis scale negative 7 to 7 by 1’s. Vertical axis scale negative 5 to 5 by 1’s. The coordinates of the triangle are (negative 1 comma negative 2), (3 comma 1), (2 comma negative 1 ). The coordinates of the image are(negative 2 comma negative 4), (6 comma 2), (4 comma negative 2).

Visual / Anchor Chart

Standards

Addressing
8.G.A

No additional information available.

8.G.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Building Toward
8.G.4

Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane.