Answer:
Part 1) A translation, followed by a dilation will map one circle onto the other, thus proving that the circles are similar
Part 2) The scale factor is equal to 1/3
Step-by-step explanation:
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. Â
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 (8,5) and has a radius of 6 units
Circle 2 is centered at (-2,1) and has a radius of 2 units
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-10,y-4)
That means----> The translation is 10 units to the left and 4 units down
so
(8,5)------> (8-10,5-4)-----> (-2,1)
center circle 1 is now equal to center circle 2 Â
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
The scale factor is equal to divide the radius of circle 2 by the radius of circle 1
scale factor=radius circle 2/radius circle 1-----> 2/6=1/3
radius circle 1 will be=6*scale factor-----> 6*(1/3)=2 units
radius circle 1 is now equal to radius circle 2 Â
therefore
A translation, followed by a dilation will map one circle onto the other, thus proving that the circles are similar
