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Lesson 4Dilations on a Square Grid

Let’s dilate figures on a rectangular grid.

Learning Targets:

  • I can apply dilations to figures on a rectangular 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.

4.1 Estimating a Scale Factor

Points A, B, and C are shown.

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. 

4.2 Dilations on a Grid

  1. Find the dilation of quadrilateral ABCD with center P and scale factor 2.  
    Rectangle ABCD is shown on a grid. Point P is within the rectangle.
  2. Find the dilation of triangle QRS with center T and scale factor 2.
  3. Find the dilation of triangle QRS with center T and scale factor \frac{1}{2} .
Triangle QRS is shown next to Point T.

4.3 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 circle is shown on a graph with Points B, C, D, H, and G.

Lesson 4 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.

Points Q, Q', and P are shown on a graph.

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 (\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 are shown on a graph. One triangle has points at (-1,-2), (3,1), and (2,-1). The other triangle has points at (-2,-4), (4,-2), and (6,2).

Lesson 4 Practice Problems

  1. Triangle ABC is dilated using D as the center of dilation with scale factor 2.

    The image is triangle A’B’C’ . Clare says the two triangles are congruent, because their angle measures are the same. Do you agree? Explain how you know.

    A larger similar triangle to Traingle ABC is shown. There is also a Point D to the left of the two triangles.

  2. On graph paper, sketch the image of quadrilateral PQRS under the following dilations:

    • The dilation centered at R with scale factor 2.
    • The dilation centered at O with scale factor \frac{1}{2} .
    • The dilation centered at S with scale factor \frac{1}{2} .
    Parallelogram PQRS is shown on a grid. There is also a Point O next to the figure.

  3. The diagram shows three lines with some marked angle measures.

    Intersecting lines create angles within two circles. One circle has an angle of 35 degrees ad the other circle has an angle of 27 degrees. The other angles are undetermined.

    Find the missing angle measures marked with question marks.

  4. Describe a sequence of translations, rotations, and reflections that takes Polygon P to Polygon Q.

    two of the same figure on a square grid in different orientations and position

  5. Point B has coordinates (\text-2,\text-5) . After a translation 4 units down, a reflection across the y -axis, and a translation 6 units up, what are the coordinates of the image?