Lesson 4Dilations on a Square Grid

Learning Goal

Let’s dilate figures on a square grid.

Learning Targets

I can apply dilations to figures on a square grid.
If I know the angle measures and side lengths of a polygon, I know the angles measures and side lengths of the polygon if I apply a dilation with a certain scale factor.

Lesson Terms

center of dilation
dilation
scale factor

Warm Up: Estimating a Scale Factor

Point $C$ is the dilation of point $B$ with center of dilation $A$ and scale factor $s$ . Estimate $s$ . Be prepared to explain your reasoning.

Activity 1: Dilations on a Grid

Problem 1

Find the dilation of quadrilateral $A B C D$ with center $P$ and scale factor 2.

A blank grid with a rectangle ABCD drawn in with a point P in the rectangle. Side length AD is 4 and side length AB is 6.

Problem 2

Find the dilation of triangle $Q R S$ with center $T$ and scale factor 2.
Find the dilation of triangle $Q R S$ with center $T$ and scale factor $\frac{1}{2}$ .

Center of dilation point T with three lines extending out and triangle QRS intersecting those lines.

Activity 2: Card Sort: Matching Dilations on a Coordinate Grid

Your teacher will give you some cards. Each of Cards 1 through 6 shows a figure in the coordinate plane and describes a dilation.

Each of Cards A through E describes the image of the dilation for one of the numbered cards.

Match number cards with letter cards. One of the number cards will not have a match. For this card, you’ll need to draw an image.

Are you ready for more?

The image of a circle under dilation is a circle when the center of the dilation is the center of the circle. What happens if the center of dilation is a point on the circle? Using center of dilation $(0, 0)$ and scale factor 1.5, dilate the circle shown on the diagram. This diagram shows some points to try dilating.

A coordinate plane with a circle drawn in and points on the circle: B (0,0), C, D, G, H. The diameter is 8 units.

Lesson Summary

Square grids can be useful for showing dilations. The grid is helpful 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, suppose we want to dilate point $Q$ with center of dilation $P$ and scale factor $\frac{3}{2}$ . Since $Q$ is 4 grid squares to the left and 2 grid squares down from $P$ , the dilation will be 6 grid squares to the left and 3 grid squares down from $P$ (can you see why?). The dilated image is marked as $Q^{'}$ in the picture.

A blank grid with points Q', Q, and P plotted.

Sometimes the square grid comes with coordinates. The coordinate grid gives us a convenient way to name points, and sometimes the coordinates of the image can be found with just arithmetic.

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

A green triangle with coordinates: (-1, -2), (3, 1), 2, -1) and a blue triangle with coordinates: (-2, -4), (6, 2), (4, -2)