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: Explain similarity of circles

All circles have the same shape. The general equation of a circle is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center and \(r\) is the radius. To map one circle \((x - a_1)^2+(y - b_1)^2=r_1^2\) onto another \((x - a_2)^2+(y - b_2)^2=r_2^2\), we first use a translation. The translation vector \(\vec{v}=\langle a_2 - a_1,b_2 - b_1
angle\) moves the center \((a_1,b_1)\) to \((a_2,b_2)\). Then, we use a dilation. The scale factor \(k = \frac{r_2}{r_1}\) (if \(r_1
eq0\)) scales the circle with radius \(r_1\) to the circle with radius \(r_2\). Since we can always find such a sequence of a translation and a dilation, all circles are similar.

Step2: Find other always - similar shapes

Spheres are always similar. The general equation of a sphere in 3 - D is \((x - a)^2+(y - b)^2+(z - c)^2 = r^2\). Similar to circles, for two spheres \((x - a_1)^2+(y - b_1)^2+(z - c_1)^2=r_1^2\) and \((x - a_2)^2+(y - b_2)^2+(z - c_2)^2=r_2^2\), we can use a translation \(\vec{v}=\langle a_2 - a_1,b_2 - b_1,c_2 - c_1
angle\) to move the center \((a_1,b_1,c_1)\) to \((a_2,b_2,c_2)\) and a dilation with scale factor \(k=\frac{r_2}{r_1}\) (if \(r_1
eq0\)) to map one sphere onto another.

Answer:

a. All circles are similar because we can always use a translation (to move the center) and a dilation (to adjust the radius) to map one circle onto another.
b. Spheres are always similar. For two spheres with equations \((x - a_1)^2+(y - b_1)^2+(z - c_1)^2=r_1^2\) and \((x - a_2)^2+(y - b_2)^2+(z - c_2)^2=r_2^2\), first translate the center \((a_1,b_1,c_1)\) to \((a_2,b_2,c_2)\) using the vector \(\langle a_2 - a_1,b_2 - b_1,c_2 - c_1
angle\), then dilate with scale factor \(k = \frac{r_2}{r_1}\) (if \(r_1
eq0\)) to map one sphere onto the other.