Question

Circle 1: center (8, 5) and radius 6

Circle 2: center (−2, 1) and radius 2
What transformations can be applied to Circle 1 to prove that the circles are similar?
What scale factor does the dilation from Circle 1 to Circle 2 have?
Show your work

Answer 1

All circles are similar to one another, and to prove this similarity no transformation other than dilation is needed. The scale factor for the dilation from Circle 1 to Circle 2 is 1/3, as the radius of Circle 2 is a third of the radius of Circle 1.

Step-by-step explanation:

To prove that Circle 1 and Circle 2 are similar, we should recognize that all circles are similar to each other, as they have the same shape and their radii are proportional. No specific transform, other than dilation, is needed to show similarity.

The scale factor for the dilation from Circle 1 to Circle 2 can be found by dividing the radius of Circle 2 by the radius of Circle 1:

Scale factor = radius of Circle 2 / radius of Circle 1 = 2 / 6 = 1/3.

Therefore, Circle 1 can be reduced to a third of its size to become Circle 2, which is a dilation with a scale factor of 1/3.

Answer 2

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.

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

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

therefore

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