All Circles are Similar (Grade 10)

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All Circles are Similar

Given two circles, one with center A (pictured below) and one with center B (not pictured), are they similar?

Circle with Center Point A
Which of the following is the best definition for similar figures?
  1. Figures which are similar in area and perimeter.
  2. Figures having congruent sides and angles.
  3. One shape can be mapped to another through a sequence of rotations, translations, reflections, and/or dilation.
  4. One shape can be mapped to another through rigid transformations.
In order to compare the two circles (circle A and B), we would need to translate one onto the other. What part of the circle should we focus on in order to make this comparison?
  1. Center
  2. Radius
  3. Circumference
  4. Chord
If we are to translate circle B to circle A, which of the following translations would accomplish this?
  1. Translate circle B by [math]vec{AB}[/math].
  2. Translate circle B by [math]vec{BA}[/math].
  3. Translate circle B by its radius.
  4. Translate circle B by the radius of circle A.
Once the center of circle B is on the center of circle A by a translation, let us assume that the radius of circle A, [math]r_A[/math], is larger than the radius of circle B, [math]r_B[/math]. What scale factor of dilation will increase [math]r_B[/math] such that the two radii will be equal?
  1. [math]r_A/r_B[/math]
  2. [math]r_B/r_A[/math]
  3. [math]r_A+r_B[/math]
  4. [math]r_A r_B[/math]
After increasing the radius of circle B, such that it is equal to the radius of circle A, how can we be sure that the two circles will lie exactly on top of each other?
  1. Since a circle is defined as a set of points a certain distance from a center, all the points on circle B will now be the same distance away from A as the points on circle A.
  2. Because a dilation is a rigid transformation, all the points on circle B must move by the same amount.
  3. Since the centers of the two circles are coincident, the rest of the circle must also be coincident.
  4. We can't be sure and need to use more radii from each circle to show this.
If circle B's radius is larger that circle A's radius, what must change in our work to show that the two circles are similar?
  1. Nothing needs to change.
  2. They cannot be similar in this case.
  3. Circle A must be translated to circle B.
  4. The dilation scale factor must change.
If the two circles' radii are congruent, can the circles still be shown to be similar?
  1. No, since they are congruent.
  2. No, since there exists no scale factor that will dilate one circle onto the other.
  3. Yes, since all congruent shapes are similar (with scale factor 1).
  4. Yes, since congruent radii means they have the same area, and any shapes with equal areas are similar.
Is the work and are conclusions from parts B through G enough to prove that any two circles are similar? If not, why?
  1. Yes, they are enough.
  2. No, we need to consider the case where we move circle A instead of circle B.
  3. No, we need set up a two-column table.
  4. No, we need to consider more sizes of circles first.
Not only are all circles similar, but all spheres are similar as well. Are these two facts related? Why or why not?
  1. No, it is just chance.
  2. No, a circle is two dimensional and a sphere is three dimensional.
  3. Yes, the reason being that both for circles and spheres, their shape is always the same and their size depends on the radius.
  4. Yes, the reason being that both are not polygons.
Given the work in steps B - H to show that two circles are similar, could we use this to show that any two spheres are similar? If so, what needs to be changed or added?
  1. No, this is not possible.
  2. Yes, by starting with spheres, and projecting them onto the xy-plane, we can then follow the steps already established.
  3. Yes, but we would need to perform completely different transformations since we are now working in three dimensions.
  4. Yes, start with spheres instead of circles and then follow nearly identical steps.

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