Question

Very Confused please help!

Answer 1

We know that
Figures can be proven similar if one, or more, similarity transformations (reflections, translations, rotations, dilations) can be found that map one figure onto another. 
In this problem to prove circle 1 and circle 2 are similar, a translation and a scale factor (from a dilation) will be found to map one circle onto another.

we have that
 Circle 1 is centered at (5,8) and has a radius of 8 centimeters
 Circle 2 is centered at (1,-2) and has a radius of 4 centimeters

step 1
Move the center of the circle 1 onto the center of the circle 2
the transformation has the following rule
(x,y)--------> (x-4,y-10)
so
(5,8)------> (5-4,8-10)-----> (1,-2)
so
center circle 1 is now equal to center circle 2 
The circles are now concentric (they have the same center)

step 2
A dilation is needed to decrease the size of circle 1 to coincide with circle 2

scale factor=radius circle 2/radius circle 1-----> 4/8----> 0.5

radius circle 1 will be=8*scale factor-----> 8*0.5-----> 4 cm
radius circle 1 is now equal to radius circle 2 

A translation, followed by a dilation will map one circle onto the other, thus proving that the circles are similar

the answer is
The circles are similar because you can translate Circle 1 using the transformation rule (x-4,y-10) and then dilate it using a scale factor of (0.5)