Lesson 2 Circle Dilations Solidify Understanding

Jump Start

Here is a circle I cut out by tracing a plate onto paper. I need to locate the center of this circle. How might I do so if I don’t want to mess it up by folding it in half?

1.

You worked on this problem as the Exit Ticket for the previous lesson. Describe your strategy for finding the center of the circle to your partner.

2.

With your partner, describe how this work might help you find the area or the circumference of this circle?

Learning Focus

Prove that all circles are similar.

It seems intuitively obvious that circles are similar, but how do we prove it?

What key features of circles are revealed by the fact that all circles are similar?

Open Up the Math: Launch, Explore, Discuss

The statement “all circles are similar” may seem reasonable, since all circles have the same shape even though they may be different sizes. However, we can learn a lot about the properties of circles by working on the proof of this statement.

Remember that the definition of similarity requires us to find a sequence of dilations and rigid motion transformations that superimposes one figure onto the other.

Liam is describing to Noah how he would prove that circle $A$ is similar to circle $B$ .

Liam: “Translate circle $A$ until its center coincides with the center of circle $B$ . Then enlarge circle $A$ by dilation until the points on circle $A$ coincide with the points on circle $B$ . Or, you could shrink circle $B$ by dilation until the points on circle $B$ coincide with the points on circle $A$ .”

Noah has some questions:

“After the translation, what is the scale factor for the enlargement that carries circle $A$ onto circle $B$ ?”
“What is the scale factor for the reduction that carries circle $B$ onto circle $A$ ?”
“How do you know the result of the dilation is still a circle?”

1.

How would you answer Noah’s questions?

Based on Liam and Noah’s discussion, we are probably convinced that circle $A$ and circle $B$ are similar. Another way we might convince ourselves that the two circles are similar would be to find the center of dilation on $\overset{\leftrightarrow}{R S}$ that maps pre-image points from circle $A$ onto corresponding image points on circle $B$ .

2.

Locate the center of dilation on $\overset{\leftrightarrow}{R S}$ that carries circle $A$ onto circle $B$ . Explain how you know the point you found is the center of dilation. (Note that both circles have been drawn tangent to $\overset{\leftrightarrow}{R S}$ .)

3.

Draw some chords, triangles, or other polygons inscribed in each circle that would be similar to each other. Explain how you know these corresponding figures are similar.

4.

Based on the figures you drew in problem 3, write some proportionality statements that you know are true.

5.

Here is a proportionality statement you may not have considered. What convinces you that it is true?

$\frac{circumference of circle A}{diameter of circle A} = \frac{circumference of circle B}{diameter of circle B}$

Since this ratio of circumference to diameter is the same scale factor for all circles, this ratio has been given the name $π$ (pi).

6.

How much larger is the circumference of circle $B$ than the circumference of circle $A$ ?

7.

Do you think the following proportion is true or false? Why?

$\frac{area of circle B}{area of circle A} = \frac{circumference of circle B}{circumference of circle A}$

Ready for More?

Develop a strategy for finding the center of dilation for two circles whose centers are not marked.

Takeaways

Our work with finding the center of a circle involved the following theorems:

All circles are similar.

One way we can demonstrate this is to translate , and then dilate by a factor of centered at .

Another way we can demonstrate this is to find the center of dilation that carries one circle onto the other. To find this center we can draw the following two lines to connect corresponding points of the two circles:

The center of dilation will be the intersection of these two lines.

The scale factor for this dilation can be found by , which will be the same as .

Measurable features of similar circles that are related by this scale factor include , , and .

The scale factor for the area of two similar circles is

Lesson Summary

In this lesson, we learned how to demonstrate that two circles are similar. One method was to translate one circle so that it coincides with the other circle. Then we can dilate the smaller circle about this common center until it coincides with the outer circle. A second method involved finding the center of dilation that would carry one circle onto the other. The formulas we use to find circumference or area of circles are dependent upon the fact that all circles are similar.

Retrieval

1.

Find the measures of angles $x$ and $y$ .

2.

Construct the line of reflection. Then write the equation of the line of reflection.

Equation: