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1. Proving the Similarity of All Circles
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Chapter 5
1. 

Proving the Similarity of All Circles

In geometry, circles hold a special place due to their unique properties. One of the most intriguing aspects is the inherent similarity of all circles, regardless of their size. This means that any two circles, when scaled appropriately, will match perfectly. This similarity is not just about size; it extends to all properties and characteristics of circles. Such proofs are fundamental in understanding the deeper concepts of geometry and have practical implications in various fields, from art to engineering. Recognizing the similarity of circles aids in simplifying complex problems and offers a clearer perspective on geometric relationships.
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Proving the Similarity of All Circles
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In this lesson, it will be proven that all circles are similar.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Try your knowledge on these topics.

a Two similar figures have the same
b Select the name of the transformation shown in each diagram.
transformation
transformation
transformation
transformation
Explore

Circle Transformation

In the diagram below, can the larger circle be mapped onto the smaller circle? What transformations are involved?

circles
Discussion

Similarity of Concentric Circles

Consider two concentric circles with common center and radii and

A pair of concentric circles with a common center O and radii of 4 and 6 units respectively.

Let and be two points on the smaller and larger circles, respectively, collinear with Also, consider the ray through and with endpoint

concentric circles

Since point is the image of point after a dilation with center and scale factor Likewise, if any point on the smaller circle is dilated through by a factor of its image will fall on the larger circle. Hence, the circles are related by a dilation, and so, are similar. The same argument can be used for any two concentric circles. This leads to the following conclusion.

Concentric circles are similar.

Concentric circles can be mapped onto each other by dilation. Therefore, concentric circles are said to be similar by dilation.
Illustration

Dilating Concentric Circles

Dilate the concentric circles below so that they map onto each other.

concentric circles
Example

Dilating a Small Circle to Map Onto a Large Circle

Given two concentric circles on a coordinate plane, Zosia is trying to determine the transformation that maps the smaller circle onto the larger circle.

concentric circles
So far, Zosia is considering four different options. Help Zosia make up her mind!

Hint

Consider the fact that the circles have different radii.

Solution

Note that the circles have different radii. The radius of the smaller circle is and the radius of the larger circle is

concentric circles
Therefore, to map the smaller circle onto the larger circle, a dilation must be performed. Furthermore, since the radius of the larger circle is twice the radius of the smaller circle, the scale factor of the dilation must be Finally, since the circles are concentric, the center of dilation is the center of the circles, which is the point
concentric circles
Example

Dilating a Large Circle to Map Onto a Small Circle

Now, given a different pair of concentric circles, Zosia wants to find the transformation that maps the larger circle onto the smaller circle.

concentric circles
Again, Zosia is considering four different options. Help Zosia make up her mind!

Hint

Consider the fact that the circles have different radii.

Solution

It can be seen in the diagram that the radius of the smaller circle is and the radius of the larger circle is

concentric circles
Therefore, to map the larger circle onto the smaller circle, a dilation must be performed. Furthermore, since the radius of the smaller circle is one third the radius of the larger circle, the scale factor of the dilation must be Finally, since the circles are concentric, the center of dilation is the center of the circles, which is the point
concentric circles
Discussion

Similarity of Circles

Now, consider two circles with different centers. One of the circles can be translated so that they become concentric circles.
concentric circles
Recall that any two concentric circles are similar through dilation.
concentric circles
Therefore, any two non-concentric circles can be mapped onto each other. This can be done either by translation or by a combination of a translation and a dilation. Since translations and dilations are similarity transformations, the following statement can be made.

Non-concentric circles are similar.

With this information and knowing that any two concentric circles are also similar, a more general statement can be made.

All circles are similar.

Illustration

Transforming Non-Concentric Circles

Two non-concentric circles can be mapped onto each other through either a translation or by a combination of a translation and a dilation. Therefore, non-concentric circles are similar. Translate and dilate the non-concentric circles below so that they map onto each other.

circles
Example

Mapping Congruent Circles Using Transformations

Diego has been asked to identify the transformation that maps the circle on the left onto the circle on the right.

non-concentric circles
Diego is considering the four different options that are shown below. Which is the correct choice?

Hint

Consider the fact that the circles have the same radius.

Solution

It can be seen in the diagram that the radius of both circles is

concentric circles
Therefore, to map the circle on the left onto the circle on the right, translation is the only transformation that must be performed. The circle on the left must be translated units to the right and units up.
concentric circles
Example

Mapping Similar Circles Using Transformations

Finally, Diego wants to identify the combination of transformations that maps the smaller circle onto the larger circle.

concentric circles
Diego is considering four different options. Help Diego make up his mind for the last time!

Hint

The radius of the smaller circle is and the radius of the larger circle is By which number does need to be multiplied to obtain

Solution

It can be seen in the diagram that the smaller circle has a radius of and its center at the point Furthermore the larger circle is centered at and its radius is

concentric circles
Therefore, to map the smaller circle onto the larger circle, a translation and a dilation must be combined. The translation must be performed units to the right and units up. Then, since is equal to the scale factor of the dilation must be
concentric circles
Closure

Extending to Theorems About Circles

In this lesson, it has been proven that all circles are similar by using similarity transformations. Therefore, any theorem that is valid for one circle, is also valid for all circles. This can be exemplified by the following theorem.

Inscribed Angles of a Circle Theorem

If two inscribed angles of a circle intercept the same arc, then they are congruent.

The diagram visualizes the theorem.
inscribed angles
This theorem will be proven later in this course.


Level 1
Level 2
Level 3
The scale factor from to is What is the circumference if is cm?

We know that the scale factor from ⊙ A to ⊙ B is 13. Let's recall that in similar figures the ratio of corresponding lengths equals the scale factor. This is true for the circumference of circles as well. Therefore, we can write the following equation. C_B/C_A= 1/3 ⇔ C_B=1/3C_A If we substitute 12 for C_A, we can find the circumference of B.

The circumference of ⊙ B is 4 cm.

Ramsha is installing a circular fountain in her backyard. In addition to the fountain, Ramsha will lay grass around the fountain.

A fountain with a grass circle on which the fountain stands.
What is the scale factor if we were to dilate the fountain's base to match the circle created by the grass? Ramsha wants the circle of grass to have a radius of feet.

We know that the radius of the grass circle is 10 feet. Notice that the radius of the grass is the sum of the lengths we see in the diagram.

With this information, we can write and solve an equation containing r. r+r+2=10 ⇔ r=4 As we can see, the radius of the fountain is 4 feet. Now that we know the radius of both circles, we can determine the scale factor between them. r_G/r_F=10/4=5/2

Consider the following circles.

6 circles of different radius.
 How many sets of concentric circles can you find?
Which of the circles are similar to

Concentric circles are circles with a common center but different radii. When we look at the diagram, we can see that i and ii have a common center but different radius which means we have found one set of concentric circles.

However, we can also see a second set of concentric circles in the diagram. Both iii. and iv. have a common center and different radii as well.

This means we have two sets of concentric circles.

According to the Similar Circles Theorem, all of the drawn circles are similar to i.

Proving the Similarity of All Circles

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