# Circle A has a radius of #4 # and a center of #(8 ,5 )#. Circle B has a radius of #2 # and a center of #(6 ,1 )#. If circle B is translated by #<2 ,7 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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To determine if circle B overlaps with circle A after being translated, calculate the distance between the centers of the circles after the translation. Then, compare this distance to the sum of the radii of the two circles. If the distance between the centers is less than the sum of the radii, the circles overlap. Otherwise, they do not overlap.

The distance formula between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is given by:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

For circle B, after translation, the new center becomes ( (6 + 2, 1 + 7) = (8, 8) ).

The distance between the centers of the circles is:

[ d = \sqrt{(8 - 8)^2 + (5 - 8)^2} = \sqrt{0 + 9} = 3 ]

The sum of the radii of circles A and B is ( 4 + 2 = 6 ).

Since the distance between the centers (3) is less than the sum of the radii (6), the circles overlap.

To find the minimum distance between points on both circles, subtract the sum of the radii from the distance between the centers:

[ \text{Minimum distance} = 3 - 6 = -3 ]

However, since the minimum distance between points on both circles cannot be negative, it indicates an overlap. Thus, the circles intersect at some points, and the minimum distance is 0.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Circle A has a radius of #1 # and a center of #(1 ,2 )#. Circle B has a radius of #2 # and a center of #(8 ,1 )#. If circle B is translated by #<-4 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- Circle A has a radius of #2 # and a center of #(7 ,3 )#. Circle B has a radius of #3 # and a center of #(2 ,2 )#. If circle B is translated by #<1 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(3 ,2 )# and #(5 ,4 )#. The line segment is dilated by a factor of #4 # around #(2 ,3 )#. What are the new endpoints and length of the line segment?
- A triangle has corners at #(3, 8 )#, ( 5, -2)#, and #( 1, -1)#. If the triangle is reflected across the x-axis, what will its new centroid be?
- Circle A has a radius of #1 # and a center of #(1 ,2 )#. Circle B has a radius of #2 # and a center of #(5 ,1 )#. If circle B is translated by #<-2 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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