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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.

Lesson Settings & Tools

	11 Theory slides
	7 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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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.

Similarity
Similarity transformations

Try your knowledge on these topics.

a Two similar figures have the same .

b Select the name of the transformation shown in each diagram.

Explore

Circle Transformation

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

Discussion

Similarity of Concentric Circles

Consider two concentric circles with common center O and radii 4 and 6.

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

Since 4* 32= 6, point B is the image of point A after a dilation with center O and scale factor 32. Likewise, if any point on the smaller circle is dilated through O by a factor of 32, 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.

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.

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 3, and the radius of the larger circle is 6.

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 2. Finally, since the circles are concentric, the center of dilation is the center of the circles, which is the point (2,- 1).

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.

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 3, and the radius of the larger circle is 9.

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 13. Finally, since the circles are concentric, the center of dilation is the center of the circles, which is the point (4,1).

Discussion

Similarity of Circles

Now, consider two circles with different centers. One of the circles can be translated so that they become concentric circles.

Recall that any two concentric circles are similar through dilation.

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.

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.

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 3.

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 9 units to the right and 4 units up.

Example

Mapping Similar Circles Using Transformations

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

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

Hint

The radius of the smaller circle is 3, and the radius of the larger circle is 4.5. By which number does 3 need to be multiplied to obtain 4.5?

Solution

It can be seen in the diagram that the smaller circle has a radius of 3 and its center at the point (- 3,- 2). Furthermore the larger circle is centered at (5.5,2.5) and its radius is 4.5

Therefore, to map the smaller circle onto the larger circle, a translation and a dilation must be combined. The translation must be performed 8.5 units to the right and 4.5 units up. Then, since 3* 1.5 is equal to 4.5, the scale factor of the dilation must be 1.5.

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.

This theorem will be proven later in this course.

Level 1

Level 2

Level 3

Proving the Similarity of All Circles

Exercise 1.1

Which of the following circles are similar to iv?

Similarity means that two figures have the same shape. Basically what this tells us is that its only their size that differs. For that to be the case, the ratio of each figure's corresponding sides must be the same. Below we see an example.

However, in circles there is only one dimension that decides the size — the radius. Therefore, we will always be able to map any two circles onto each other by dilating the radius of one of them. It is formalized by the Similar Circles Theorem that all circles are similar.

Is there enough information to claim that ⊙ A ~ ⊙ B. Explain your reasoning.

The symbol ~ means similarity. Therefore, the statement is telling us that circle A is similar to circle B. A ~ B ⇓ A is similar to B By the Similar Circles Theorem, we know that all circles are, in fact, similar. We can prove this by transforming one of the circles until it maps onto the other. In this case, the circles have the same radius. Therefore, all we need to do is translate one circle until the centers overlap.

Consider ⊙ A and ⊙ B.

By which scale factor do you have to dilate ⊙ A to make it map onto ⊙ B?

By which scale factor do you have to dilate ⊙ B to make it map onto ⊙ A.

The scale factor needs to dilate ⊙ A such that it maps onto ⊙ B is the ratio of the radius of A to the radius of B. Therefore, let's first measure the radii of the two circles.

As we can see, ⊙ A and ⊙ B has a radius of 5 and 2 units respectively. Therefore, to dilate A so that it maps onto ⊙ B, we have to multiply its radius by a scale factor of 25.

This time, we want to dilate ⊙ B until it maps onto ⊙ A. From Part A, we know the radii of both circles. Since we want B to map onto A, we must multiply the radius of B by a scale factor of 52.

Proving the Similarity of All Circles

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