# A has center of (0, 4) and a radius of 6, and circle B has a center of (-3, 5) and a radius of 24. What steps will help show that circle A is similar to circle B?

Jul 6, 2016

Two circles are similar because they have the same shape: they are both circles.
It's difficult to see what this question is asking. Perhaps the Explanation below is what is being looked for.

#### Explanation:

Two shapes are similar if by applying

• translation (shift);
• dilation (resizing);
• reflection; and/or
• rotation

they can be mapped into identical forms.

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Note that

• Circle A with center (""(0,4)) and radius $6$
has a Cartesian equation
$\textcolor{w h i t e}{\text{XXX}} {\left(x - 0\right)}^{2} + {\left(y - 4\right)}^{2} = {6}^{2}$
• Circle B with center ("(-3,5)) and radius $24$
has a Cartesian equation
color(white)("XXX")(x+3)^2+(y-5)^2=24^2

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In the Cartesian plane:

translation (shift) of color(black)(vec(""(a,b))
Every coordinate $x$ value has $a$ added to it; and
every coordinate $y$ value has $b$ added to it.

Applications
Applying a translation of vec(""(color(red)(0),color(blue)(4))) to circle A
gives the circle A' with the equation:
$\textcolor{w h i t e}{\text{XXX}} {\left(\left(x \textcolor{red}{0}\right) - 0\right)}^{2} + {\left(\left(y \textcolor{b l u e}{+ 4}\right) - 4\right)}^{2} = {6}^{2}$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {x}^{2} + {y}^{2} = {6}^{2}$
Applying a translation of vec(""(color(red)(3),color(blue)(-5)))# to circle B
gives a circle B' with the equation:
$\textcolor{w h i t e}{\text{XXX}} {\left(\left(x \textcolor{red}{+ 3}\right) - 3\right)}^{2} + {\left(\left(y \textcolor{b l u e}{- 5}\right) + 5\right)}^{2} = {24}^{2}$

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dilation of $\textcolor{b l a c k}{d}$
Distances from the origin are multiplied by $\textcolor{b l a c k}{d}$

Application
(note for the equation of a circle, only the radius is a $\underline{\text{distance}}$ measurement).
Applying a dilation of $4$ to the equation of Circle A'
gives the circle A'' with equation
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} + {y}^{2} = {\left(6 \times 4\right)}^{2}$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {x}^{2} + {y}^{2} = {24}^{2}$

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Since the equations for A'' and B' are identical
and
Circle A'' is the same as Circle A after translation and dilation
and
Circle B' is the same as Circle B after translation.

$\Rightarrow$ Circle A and Circle B are similar.