Question

3 - 55. remember that two figures are similar whenever there is a sequence of transformations (including dilation) that carries one onto the other.
a. explain why all circles must be similar. that is, describe a sequence of transformations that will always carries one circle onto another.
b. can you think of any other shapes that are always similar? if you can, draw an example and explain why they are always similar.

Explanation:

Step1: Similarity of circles

All circles have the same shape. The only difference between circles is their size (determined by the radius). A dilation (a transformation that changes the size of a figure) can be used to make one circle the same size as another. Also, we can use translation (to move the circle to the position of the other circle). So, we can always map one circle onto another using a dilation and a translation.
Let the first circle have center \((x_1,y_1)\) and radius \(r_1\), and the second circle have center \((x_2,y_2)\) and radius \(r_2\).
First, perform a translation \(T\) that maps the center \((x_1,y_1)\) to \((x_2,y_2)\). The translation rule is \(T(x,y)=(x-(x_1 - x_2),y-(y_1 - y_2))\).
Then, perform a dilation \(D\) with scale factor \(k=\frac{r_2}{r_1}\). The dilation rule is \(D(x,y)=(kx,ky)\)

Step2: Other always - similar shapes

Equilateral triangles are always similar.
An equilateral triangle has all sides equal and all angles equal to \(60^{\circ}\). Let \(\triangle ABC\) and \(\triangle DEF\) be two equilateral triangles.
First, use a translation to move one triangle so that one of its vertices coincides with a vertex of the other triangle. Then, since the ratio of the side lengths of two equilateral triangles is a constant (if the side lengths are \(a\) and \(b\), the scale factor of dilation \(k = \frac{b}{a}\)), a dilation can be used to make the two equilateral triangles congruent (and thus similar, as congruent figures are a special case of similar figures with a scale factor of \(1\)).

Answer:

a. All circles are similar because we can use a translation (to move the center of one circle to the center of another) and a dilation (to adjust the size) to map one circle onto another.
b. Equilateral triangles are always similar.