Lesson 2

Circular Grid

Let’s dilate figures on circular grids.

2.1: Notice and Wonder: Concentric Circles

Blank circular grid.

What do you notice? What do you wonder?

2.2: A Droplet on the Surface

The larger Circle d is a dilation of the smaller Circle c. \(P\) is the center of dilation.

  1. Draw four points on the smaller circle using the Point on Object tool.

    geogebra point on object tool

  2. Draw the rays from \(P\) through each of those four points. Select the Ray tool, then point \(P\), and then the second point.

    geogebra select ray tool

  3. Mark the intersection points of the rays and Circle d by selecting the Intersect tool and clicking on the point of intersection.

    geogebra select intersection tool

     

  4. Complete the table. In the row labeled S, write the distance between \(P\) and the point on the smaller circle in grid units. In the row labeled L, write the distance between \(P\) and the corresponding point on the larger circle in grid units. Measure the distances between pairs of points by selecting the Distance tool, and then clicking on the two points.

    geogebra distance tool
      \(A\) \(B\) \(C\) \(D\)
    S        
    L        
  5. The center of dilation is point \(P\). What is the scale factor that takes the smaller circle to the larger circle? Explain your reasoning.

2.3: Quadrilateral on a Circular Grid

 Here is a polygon \(ABCD\)

  1. Dilate each vertex of polygon \(ABCD\) using \(P\) as the center of dilation and a scale factor of 2. 

     
  2. Draw segments between the dilated points to create a new polygon.

  3. What are some things you notice about the new polygon?

  4. Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?

  5. Dilate each vertex of polygon \(ABCD\) using \(P\) as the center of dilation and a scale factor of \(\frac{1}{2}\).

  6. What do you notice about this new polygon?



Suppose \(P\) is a point not on line segment \(\overline{WX}\). Let \(\overline{YZ}\) be the dilation of line segment \(\overline{WX}\) using \(P\) as the center with scale factor 2. Experiment using a circular grid to make predictions about whether each of the following statements must be true, might be true, or must be false.

  1. \(\overline{YZ}\) is twice as long \(\overline{WX}\).
  2. \(\overline{YZ}\) is five units longer than \(\overline{WX}\).
  3. The point \(P\) is on \(\overline{YZ}\).
  4. \(\overline{YZ}\) and \(\overline{WX}\) intersect.

2.4: A Quadrilateral and Concentric Circles

  1. Dilate polygon \(EFGH\) using \(Q\) as the center of dilation and a scale factor of \(\frac13\). The image of \(F\) is already shown on the diagram. (You may need to draw more rays from \(Q\) in order to find the images of other points.)

Summary

A circular grid like this one can be helpful for performing dilations.

The radius of the smallest circle is one unit, and the radius of each successive circle is one unit more than the previous one.

Blank circular grid.

To perform a dilation, we need a center of dilation, a scale factor, and a point to dilate. In the picture, \(P\) is the center of dilation. With a scale factor of 2, each point stays on the same ray from \(P\), but its distance from \(P\) doubles:

Point P, triangle A B C and its image triangle A prime B prime and C prime.

Since the circles on the grid are the same distance apart, segment \(PA'\) has twice the length of segment \(PA\), and the same holds for the other points. 

Glossary Entries

  • center of a dilation

    The center of a dilation is a fixed point on a plane. It is the starting point from which we measure distances in a dilation.

    In this diagram, point \(P\) is the center of the dilation.

    A dilation
  • dilation

    A dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point. The fixed point is the center of the dilation. All of the original distances are multiplied by the same scale factor.

    For example, triangle \(DEF\) is a dilation of triangle \(ABC\). The center of dilation is \(O\) and the scale factor is 3.

    This means that every point of triangle \(DEF\) is 3 times as far from \(O\) as every corresponding point of triangle \(ABC\).

    2 triangles. Triangle DEF is a dilation of triangle ABC.
  • scale factor

    To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.

    In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).

    2 triangles