2.1.2: Circular Grid

Lesson

Let's dilate figures on circular grids.

Exercise \(\PageIndex{1}\): Notice and Wonder: Concentric Circles

Figure \(\PageIndex{1}\)

What do you notice? What do you wonder?

Exercise \(\PageIndex{2}\): 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.
   
2. Draw the rays from \(P\) through each of those four points. Select the Ray tool, then point \(P\), and then the second point.
   
3. Mark the intersection points of the rays and Circle d by selecting the Intersect tool and clicking on the point of intersection.
   
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.
   

Table \(\PageIndex{1}\)

	\(A\)	\(B\)	\(C\)	\(D\)
\(S\)
\(L\)

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.

Exercise \(\PageIndex{3}\): Quadrilateral on a Circular Grid

Here is a polygon \(ABCD\)

1. Dilate each vertec 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?

Are you ready for more?

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.

Exercise \(\PageIndex{4}\): A Quadrilateral and Concentric Circles

1. Dilate polygon \(EFGH\) using \(Q\) as the center of dilation and a scale factor of \(\frac{1}{3}\). 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.

Figure \(\PageIndex{2}\)

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:

Figure \(\PageIndex{3}\)

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

Definition: 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.

Figure \(\PageIndex{4}\)

Definition: 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 \(ABC\).

Figure \(\PageIndex{5}\)

Definition: 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 \(4\cdot (1.5)=6\), \(5\cdot (1.5)=7.5\), and \(6\cdot (1.5)=9\).

Figure \(\PageIndex{6}\)

Practice

Exercise \(\PageIndex{5}\)

Here are Circles \(c\) and \(d\). Point \(O\) is the center of dilation, and the dilation takes Circle \(c\) to Circle \(d\).

Figure \(\PageIndex{7}\)

1. Plot a point on Circle \(c\). Label the point \(P\). Plot where \(P\) goes when the dilation is applied.
2. Plot a point on Circle \(d\). Label the point \(Q\). Plot a point that the dilation takes to \(Q\).

Exercise \(\PageIndex{6}\)

Here is a triangle \(ABC\).

Figure \(\PageIndex{8}\)

1. Dilate each vertex of triangle \(ABC\) using \(P\) as the center of dilation and a scale factor of 2. Draw the triangle connecting the three new points.
2. Dilate each vertex of triangle \(ABC\) using \(P\) as the center of dilation and a scale factor of \(\frac{1}{2}\). Draw the triangle connecting the three new points.
3. Measure the longest side of each of the three triangles. What do you notice?
4. Measure the angles of each triangle. What do you notice?

Exercise \(\PageIndex{7}\)

Describe a rigid transformation that you could use to show the polygons are congruent.

Figure \(\PageIndex{9}\)

(From Unit 1.3.2)

Exercise \(\PageIndex{8}\)

The line has been partitioned into three angles.

Figure \(\PageIndex{10}\): A straight line with two rays coming out of a single point. One slanting up and to the left. One slanting up and to the right. Three angles are formed. 39 degrees. 99 degrees. 42 degrees.

Is there a triangle with these three angle measures? Explain.

(From Unit 1.4.2)