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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 $whatever .$

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

{"type":"choice","form":{"alts":["Translation","Rotation","Reflection","Dilation"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

{"type":"choice","form":{"alts":["Rotation","Translation","Dilation","Reflection"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

{"type":"choice","form":{"alts":["Reflection","Rotation","Translation","Dilation"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

{"type":"choice","form":{"alts":["Reflection","Rotation","Translation","Dilation"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":3}

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

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 $4×23 =6,$ point B is the image of point A after a dilation with center O and scale factor $23 .$ Likewise, if any point on the smaller circle is dilated through O by a factor of $23 ,$ 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.

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

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!{"type":"choice","form":{"alts":["Dilation with center <span class=\"mlmath-simple\">(2<span class=\"space after\">,<\/span><span class=\"text\">-<\/span>1)<\/span> and scale factor <span class=\"mlmath-simple\">2.<\/span>","Dilation with center <span class=\"mlmath-simple\">(2<span class=\"space after\">,<\/span><span class=\"text\">-<\/span>1)<\/span> and scale factor <span class=\"mlmath-simple\">0.5.<\/span>","Dilation with center <span class=\"mlmath-simple\">(0<span class=\"space after\">,<\/span>0)<\/span> and scale factor <span class=\"mlmath-simple\">2.<\/span>","Translation <span class=\"mlmath-simple\">2<\/span> units up and <span class=\"mlmath-simple\">2<\/span> units to the left."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

Consider the fact that the circles have different radii.

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).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!{"type":"choice","form":{"alts":["Dilation with center <span class=\"mlmath-simple\">(4<span class=\"space after\">,<\/span>1)<\/span> and scale factor <span class=\"mlmath-simple\">3.<\/span>","Dilation with center <span class=\"mlmath-simple\">(4<span class=\"space after\">,<\/span>1)<\/span> and scale factor <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.190108em;vertical-align:-0.345em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","Dilation with center <span class=\"mlmath-simple\">(0<span class=\"space after\">,<\/span>0)<\/span> and scale factor <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.190108em;vertical-align:-0.345em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","Translation <span class=\"mlmath-simple\">6<\/span> units up and <span class=\"mlmath-simple\">3<\/span> units right."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}

Consider the fact that the circles have different radii.

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 $31 .$ Finally, since the circles are concentric, the center of dilation is the center of the circles, which is the point (4,1).
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.

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.

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?{"type":"choice","form":{"alts":["Translation <span class=\"mlmath-simple\">9<\/span> units to the left and <span class=\"mlmath-simple\">4<\/span> units down.","Translation <span class=\"mlmath-simple\">9<\/span> units to the right and <span class=\"mlmath-simple\">4<\/span> units up.","Dilation with center <span class=\"mlmath-simple\">(0<span class=\"space after\">,<\/span>0)<\/span> and scale factor <span class=\"mlmath-simple\">2.<\/span>","Translation <span class=\"mlmath-simple\">9<\/span> units to the right and <span class=\"mlmath-simple\">4<\/span> units down."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}

Consider the fact that the circles have the same radius.

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.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!{"type":"choice","form":{"alts":["Translation <span class=\"mlmath-simple\">8.5<\/span> units to the right and <span class=\"mlmath-simple\">4.5<\/span> units up, and dilation by a scale factor of <span class=\"mlmath-simple\">1.5.<\/span>","Translation <span class=\"mlmath-simple\">8.5<\/span> units to the right and <span class=\"mlmath-simple\">4.5<\/span> units up, and dilation by a scale factor of <span class=\"mlmath-simple\">2.<\/span>","Translation <span class=\"mlmath-simple\">8<\/span> units to the right and <span class=\"mlmath-simple\">4.5<\/span> units up, and dilation by a scale factor of <span class=\"mlmath-simple\">1.5.<\/span>","Translation <span class=\"mlmath-simple\">8.5<\/span> units to the right and <span class=\"mlmath-simple\">4<\/span> units up, and dilation by a scale factor of <span class=\"mlmath-simple\">1.5.<\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

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?

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.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.

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

This theorem will be proven later in this course.

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