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Circle Similarity

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

  1. What is the ratio between the radii of the circles?
  2. What is the scale factor between the two circles?
  3. What is the ratio between the circumferences of the circles?
  4. What is the ratio between the areas of the circles?
  5. What do area ratios and circumference ratios have to do with scale factor?

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Guidance

A circle is a set of points equidistant from a given point. The radius of a circle,  r , 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: C=2 \pi r
  • Area of a Circle: A=\pi r^2

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 A , centered at point  A with radius r_A , and circle D , centered at point  D with radius r_D . Perform a rigid transformation to bring point  A to point D .

Solution: Draw a vector from point  A to point D . Translate circle  A along the vector to create circle  A^\prime . Note that r_A \cong r_A^\prime .

Example B

Dilate circle  A to map it to circle D . 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  A^\prime about point  A^\prime by a scale factor of \frac{r_D}{r_{A^{\prime}}} .

Circle  A^{\prime \prime} is the same circle as circle D . You can be confident of this because  r_{A^{\prime \prime}}=\frac{r_D}{r_{A^{\prime}}} \cdot r_{A^{\prime}}=r_D and point  A^{\prime \prime} is the same as point D . 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  A is similar to circle D , because a similarity transformation (translation then dilation) mapped circle  A to circle D .

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

Example C

Show that circle  A with center  (-3, 4) and radius 2 is similar to circle  B with center  (3, 2) and radius 4.

Solution: Translate circle  A along the vector from  (-3, 4) to (3, 2) . Then, dilate the image about its center by a scale factor of 2. You will have mapped circle  A to circle  B with a similarity transformation. This means that circle  A is similar to circle B .

Concept Problem Revisited

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

  1. What is the ratio between the radii of the circles?
  2. What is the scale factor between the two circles?
  3. What is the ratio between the circumferences of the circles?
  4. What is the ratio between the area of the circles?
  5. What do area ratios and circumference ratios have to do with scale factor?

a. The ratio between the radii is \frac{3}{1} .

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  \frac{3}{1} so the scale factor is \frac{3}{1}=3 .

c. The circumference of the smaller circle is C=2 \pi (1)=2 \pi . The circumference of the larger circle is C=2 \pi (3)=6 \pi . The ratio between the circumferences is \frac{6 \pi}{2 \pi}=\frac{3}{1} .

d. The area of the smaller circle is A=\pi(1)^2=\pi . The area of the larger circle is A=\pi(3)^2=9 \pi . The ratio between the areas is \frac{9 \pi}{\pi}=\frac{9}{1} . Note that \frac{9}{1}=\left(\frac{3}{1} \right)^2 .

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

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

dilation is an example of a transformation that moves each point along a ray through the point emanating from a fixed center point P , multiplying the distance from the center point by a common scale factor, k .

Two figures are  similar if a similarity transformation will carry one figure to the other.  Similar figures will always have corresponding angles congruent and corresponding sides proportional.

circle is the set of points equidistant from a given point.

The  radius of a circle, r , is the distance from the center of the circle to the circle.

Guided Practice

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

Answers:

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

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  A with center  (2, 7) and radius 4. Circle  B with center  (1, -4) and radius 3.

2. Circle  A with center  (6, 4) and radius 3. Circle  B with center  (-5, 6) and radius 5.

3. Circle  A with center  (1, 4) and radius 2. Circle  B with center  (3, -2) and radius 7.

4. Circle  A with center  (8, 1) and radius 6. Circle  B with center  (6, -4) and radius 8.

5. Circle  A with center  (-2, 10) and radius 3. Circle  B with center  (-1, -4) and radius 6.

6. Circle  A with center  (-1, 5) and radius 4. Circle  B with center  (-1, 5) and radius 5.

7. Circle  A with center  (-4, -2) and radius 1. Circle  B with center  (1, 8) and radius 4.

8. Circle  A with center  (10, 3) and radius 5. Circle  B with center  (4, 2) and radius 8.

9. Circle  A with center  (12, 4) and radius 10. Circle  B with center  (12, 4) and radius 12.

10. Circle  A with center  (-7, 6) and radius 9. Circle  B with center  (1, -4) and radius 9.

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

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

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

14. The ratio of the area of circle  A to the area of circle  B is \frac{12}{5} . 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?

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