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?

1 Answer
Jul 6, 2016

Answer:

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
    #color(white)("XXX")(x-0)^2+(y-4)^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

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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:
#color(white)("XXX")((xcolor(red)(0))-0)^2+((ycolor(blue)(+4))-4)^2=6^2#
#color(white)("XXX")rarr 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:
#color(white)("XXX")((xcolor(red)(+3))-3)^2+((ycolor(blue)(-5))+5)^2=24^2#

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dilation of #color(black)(d)#
Distances from the origin are multiplied by #color(black)(d)#

Application
(note for the equation of a circle, only the radius is a #underline("distance")# measurement).
Applying a dilation of #4# to the equation of Circle A'
gives the circle A'' with equation
#color(white)("XXX")x^2+y^2=(6xx4)^2#
#color(white)("XXX")rarr x^2+y^2=24^2#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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.

#rArr# Circle A and Circle B are similar.