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

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 $(0,0)$ and scale factor 2 on the triangle with coordinates $(\text-1, \text-2)$, $(3,1)$, and $(2, \text-1)$, we can just double the coordinates to get $(\text-2, \text-4)$, $(6,2)$, and $(4, \text-2)$.

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).