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,
, is the distance from the center of the circle to the circle. A circle's size depends
*
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:
- Area of a Circle:

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 , centered at point with radius , and circle , centered at point with radius . Perform a rigid transformation to bring point to point .

**
Solution:
**
Draw a vector from point
to point
. Translate circle
along the vector to create circle
. Note that
.

**
Example B
**

Dilate circle to map it to circle . Can you be confident that the circles are similar?

**
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
about point
by a scale factor of
.

Circle is the same circle as circle . You can be confident of this because and point is the same as point . Because a circle is defined by its center and radius, if two circles have the same center and radius then they are the same circle.

This means that circle is similar to circle , because a similarity transformation (translation then dilation) mapped circle to circle .

Circle and circle were two random circles. This proves that in general, all circles are similar.

**
Example C
**

Show that circle with center and radius 2 is similar to circle with center and radius 4.

**
Solution:
**
Translate circle
along the vector from
to
. Then, dilate the image about its center by a scale factor of 2. You will have mapped circle
to circle
with a similarity transformation. This means that circle
is similar to 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 so the scale factor is .

c. The circumference of the smaller circle is . The circumference of the larger circle is . The ratio between the circumferences is .

d. The area of the smaller circle is . The area of the larger circle is . The ratio between the areas is . Note that .

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
, multiplying the distance from the center point by a common scale factor,
.

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,
, is the distance from the center of the circle to the circle.

#### Guided Practice

- Show that circle with center and radius 2 is similar to circle with center and radius 3.
- The ratio of the circumference of circle to the circumference of circle is . What is the ratio of their areas?
- The ratio of the area of circle to the area of circle is . What is the ratio of their radii?

**
Answers:
**

- Translate circle along the vector from to . Then, dilate the image about its center with a scale factor of . You will have mapped circle to circle with a similarity transformation. This means that circle is similar to circle .
- The ratio of the circumferences is the same as the scale factor. Therefore, the scale factor is . The ratio of the areas is the scale factor squared. Therefore, the ratio of the areas is .
- The ratio of the areas is the scale factor squared. Therefore, the scale factor is . The ratio of the radii is the same as the scale factor, so the ratio of the radii is .

#### 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 with center and radius 4. Circle with center and radius 3.

2. Circle with center and radius 3. Circle with center and radius 5.

3. Circle with center and radius 2. Circle with center and radius 7.

4. Circle with center and radius 6. Circle with center and radius 8.

5. Circle with center and radius 3. Circle with center and radius 6.

6. Circle with center and radius 4. Circle with center and radius 5.

7. Circle with center and radius 1. Circle with center and radius 4.

8. Circle with center and radius 5. Circle with center and radius 8.

9. Circle with center and radius 10. Circle with center and radius 12.

10. Circle with center and radius 9. Circle with center and radius 9.

11. The ratio of the circumference of circle to the circumference of circle is . What is the ratio of their radii?

12. The ratio of the area of circle to the area of circle is . What is the ratio of their radii?

13. The ratio of the radius of circle to the radius of circle is . What is the ratio of their areas?

14. The ratio of the area of circle to the area of circle is . 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?