Sean has two circles, one with a radius of 1 inch and another with a radius of 3 inches.

- What is the ratio between the radii of the circles?
- What is the scale factor between the two circles?
- What is the ratio between the circumferences of the circles?
- What is the ratio between the areas of the circles?
- What do area ratios and circumference ratios have to do with scale factor?

#### Watch This

https://www.youtube.com/watch?v=SSsdsff5QBg

#### Guidance

A **circle** is a set of points equidistant from a given point. The **radius** of a circle, *only* on its radius.

Two figures are **similar** if a similarity transformation will carry one figure to the other. A **similarity** **transformation** is one or more rigid transformations followed by a dilation. In the examples, you will show that a similarity transformation exists between any two circles and therefore, all circles are similar.

Recall two important formulas related to circles:

- Circumference (Perimeter) of a Circle:
C=2πr - Area of a Circle:
A=πr2

Once you have shown that all circles are similar, you will explore how the circumferences and areas of circles are related.

**Example A**

Consider circle

**Solution:** Draw a vector from point

**Example B**

Dilate circle

**Solution:** Because the size of a circle is completely determined by its radius, you can use the radii to find the correct scale factor. Dilate circle

Circle

This means that circle

Circle

**Example C**

Show that circle

**Solution:** Translate circle

**Concept Problem Revisited**

Sean has two circles, one with a radius of 1 inch and another with a radius of 3 inches.

- What is the ratio between the radii of the circles?
- What is the scale factor between the two circles?
- What is the ratio between the circumferences of the circles?
- What is the ratio between the area of the circles?
- What do area ratios and circumference ratios have to do with scale factor?

a. The ratio between the radii is

b. A scale factor exists because any two circles are similar. You can use the radii to determine the scale factor. The ratio between the radii is

c. The circumference of the smaller circle is

d. The area of the smaller circle is

e. The area ratio is the scale factor squared, because area is a two dimensional measurement. The circumference ratio is equal to the scale factor, because circumference is a one dimensional measurement.

#### Vocabulary

A ** similarity transformation** is one or more rigid transformations followed by a dilation.

A ** dilation** is an example of a transformation that moves each point along a ray through the point emanating from a fixed center point \begin{align*}P\end{align*}, multiplying the distance from the center point by a common scale factor, \begin{align*}k\end{align*}.

Two figures are ** similar** if a similarity transformation will carry one figure to the other.

**will always have corresponding angles congruent and corresponding sides proportional.**

*Similar figures*
A ** circle** is the set of points equidistant from a given point.

The ** radius** of a circle, \begin{align*}r\end{align*}, is the distance from the center of the circle to the circle.

#### Guided Practice

- Show that circle \begin{align*}A\end{align*} with center \begin{align*}(-1, 7)\end{align*} and radius 2 is similar to circle \begin{align*}B\end{align*} with center \begin{align*}(4, 6)\end{align*} and radius 3.
- The ratio of the circumference of circle \begin{align*}D\end{align*} to the circumference of circle \begin{align*}C\end{align*} is \begin{align*}\frac{4}{3}\end{align*}. What is the ratio of their areas?
- The ratio of the area of circle \begin{align*}F\end{align*} to the area of circle \begin{align*}E\end{align*} is \begin{align*}\frac{9}{4}\end{align*}. What is the ratio of their radii?

**Answers:**

- Translate circle \begin{align*}A\end{align*} along the vector from \begin{align*}(-1, 7)\end{align*} to \begin{align*}(4, 6)\end{align*}. Then, dilate the image about its center with a scale factor of \begin{align*}\frac{3}{2}\end{align*}. You will have mapped circle \begin{align*}A\end{align*} to circle \begin{align*}B\end{align*} with a similarity transformation. This means that circle \begin{align*}A\end{align*} is similar to circle \begin{align*}B\end{align*}.
- The ratio of the circumferences is the same as the scale factor. Therefore, the scale factor is \begin{align*}\frac{4}{3}\end{align*}. The ratio of the areas is the scale factor squared. Therefore, the ratio of the areas is \begin{align*}\left(\frac{4}{3} \right)^2=\frac{16}{9}\end{align*}.
- The ratio of the areas is the scale factor squared. Therefore, the scale factor is \begin{align*}\sqrt{\frac{9}{4}}=\frac{3}{2}\end{align*}. The ratio of the radii is the same as the scale factor, so the ratio of the radii is \begin{align*}\frac{3}{2}\end{align*}.

#### Practice

For #1-#10, show that the circles are similar by describing the similarity transformation necessary to map one circle onto the other.

1. Circle \begin{align*}A\end{align*} with center \begin{align*}(2, 7)\end{align*} and radius 4. Circle \begin{align*}B\end{align*} with center \begin{align*}(1, -4)\end{align*} and radius 3.

2. Circle \begin{align*}A\end{align*} with center \begin{align*}(6, 4)\end{align*} and radius 3. Circle \begin{align*}B\end{align*} with center \begin{align*}(-5, 6)\end{align*} and radius 5.

3. Circle \begin{align*}A\end{align*} with center \begin{align*}(1, 4)\end{align*} and radius 2. Circle \begin{align*}B\end{align*} with center \begin{align*}(3, -2)\end{align*} and radius 7.

4. Circle \begin{align*}A\end{align*} with center \begin{align*}(8, 1)\end{align*} and radius 6. Circle \begin{align*}B\end{align*} with center \begin{align*}(6, -4)\end{align*} and radius 8.

5. Circle \begin{align*}A\end{align*} with center \begin{align*}(-2, 10)\end{align*} and radius 3. Circle \begin{align*}B\end{align*} with center \begin{align*}(-1, -4)\end{align*} and radius 6.

6. Circle \begin{align*}A\end{align*} with center \begin{align*}(-1, 5)\end{align*} and radius 4. Circle \begin{align*}B\end{align*} with center \begin{align*}(-1, 5)\end{align*} and radius 5.

7. Circle \begin{align*}A\end{align*} with center \begin{align*}(-4, -2)\end{align*} and radius 1. Circle \begin{align*}B\end{align*} with center \begin{align*}(1, 8)\end{align*} and radius 4.

8. Circle \begin{align*}A\end{align*} with center \begin{align*}(10, 3)\end{align*} and radius 5. Circle \begin{align*}B\end{align*} with center \begin{align*}(4, 2)\end{align*} and radius 8.

9. Circle \begin{align*}A\end{align*} with center \begin{align*}(12, 4)\end{align*} and radius 10. Circle \begin{align*}B\end{align*} with center \begin{align*}(12, 4)\end{align*} and radius 12.

10. Circle \begin{align*}A\end{align*} with center \begin{align*}(-7, 6)\end{align*} and radius 9. Circle \begin{align*}B\end{align*} with center \begin{align*}(1, -4)\end{align*} and radius 9.

11. The ratio of the circumference of circle \begin{align*}A\end{align*} to the circumference of circle \begin{align*}B\end{align*} is \begin{align*}\frac{2}{3}\end{align*}. What is the ratio of their radii?

12. The ratio of the area of circle \begin{align*}A\end{align*} to the area of circle \begin{align*}B\end{align*} is \begin{align*}\frac{6}{1}\end{align*}. What is the ratio of their radii?

13. The ratio of the radius of circle \begin{align*}A\end{align*} to the radius of circle \begin{align*}B\end{align*} is \begin{align*}\frac{5}{9}\end{align*}. What is the ratio of their areas?

14. The ratio of the area of circle \begin{align*}A\end{align*} to the area of circle \begin{align*}B\end{align*} is \begin{align*}\frac{12}{5}\end{align*}. What is the ratio of their circumferences?

15. To show that any two circles are similar you need to perform a translation and/or a dilation. Why won't you ever need to use reflections or rotations?