# Circle A has a radius of #1 # and a center at #(3 ,3 )#. Circle B has a radius of #3 # and a center at #(6 ,4 )#. If circle B is translated by #<-3 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Yes they do, because the distance between the two circle centres

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The translated center of circle B after the translation is (3, 8). To find if circle B overlaps circle A, we need to calculate the distance between the centers of the two circles and compare it to the sum of their radii.

The distance between the centers of circles A and B after the translation is: [ \sqrt{(3 - 3)^2 + (3 - 8)^2} = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5 ]

The sum of the radii of circles A and B is: [ 1 + 3 = 4 ]

Since the distance between the centers of the circles (5) is greater than the sum of their radii (4), the circles do not overlap.

To find the minimum distance between points on both circles, we subtract the radii of circle A and circle B from the distance between their centers: [ \text{Minimum distance} = 5 - (1 + 3) = 5 - 4 = 1 ]

So, the minimum distance between points on both circles is 1.

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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.

- A triangle has corners at #(-4 ,2 )#, #(7 ,1 )#, and #(2 ,-7 )#. If the triangle is dilated by a factor of #2/5 # about point #(-1 ,4 ), how far will its centroid move?
- Point A is at #(2 ,-1 )# and point B is at #(9 ,-7 )#. Point A is rotated #(3pi)/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?
- Circle A has a radius of #3 # and a center at #(1 ,2 )#. Circle B has a radius of #5 # and a center at #(3 ,7 )#. If circle B is translated by #<2 ,1 >#, 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 #(2 ,5 )#. Circle B has a radius of #3 # and a center of #(3 ,8 )#. If circle B is translated by #<4 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A triangle has corners at #(4 ,2 )#, #(2 ,-3 )#, and #(8 ,-2 )#. If the triangle is dilated by a factor of #1/3 # about point #(-3 ,-1 ), how far will its centroid move?

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