# Circle A has a radius of #2 # and a center of #(6 ,2 )#. Circle B has a radius of #3 # and a center of #(5 ,7 )#. If circle B is translated by #<-1 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

no overlap , d ≈ 2.28

A translation does not change the shape of a figure , only it's position.

Centre of B (5 , 7) → (5-1 , 7 + 2 ) → (4 , 9)

now, radius of A + radius of B = 2 + 3 = 5

Since sum of radii < distance between centres , no overlap

and distance between circles (d) = 7.28 - 5 = 2.28

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To determine if circle B, after being translated, overlaps with circle A, we need to calculate the distance between the centers of the circles and compare it to the sum of their radii. If the distance between the centers is less than the sum of the radii, then the circles overlap. Otherwise, they don't overlap, and we need to find the minimum distance between points on both circles.

First, let's find the distance between the centers of the circles:

The distance formula between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is given by: [ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

For circle A with center (6, 2) and circle B after translation with center (5 - 1, 7 + 2) = (4, 9): [ \text{distance} = \sqrt{(4 - 6)^2 + (9 - 2)^2} = \sqrt{(-2)^2 + 7^2} = \sqrt{53} ]

The sum of the radii of circles A and B is ( 2 + 3 = 5 ).

Since ( \sqrt{53} > 5 ), the circles do not overlap.

To find the minimum distance between points on both circles, we subtract the sum of the radii from the distance between the centers: [ \text{minimum distance} = \sqrt{53} - 5 = \sqrt{53} - 5 ]

Therefore, the minimum distance between points on both circles is ( \sqrt{53} - 5 ).

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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 line segment has endpoints at #(3 ,7 )# and #(5 ,9)#. If the line segment is rotated about the origin by #(pi )/2 #, translated vertically by #2 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- Points A and B are at #(3 ,8 )# and #(7 ,3 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #5 #. If point A is now at point B, what are the coordinates of point C?
- Point A is at #(9 ,3 )# and point B is at #(5 ,-6 )#. Point A is rotated #pi/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 #5 # and a center of #(3 ,2 )#. Circle B has a radius of #3 # and a center of #(1 ,4 )#. 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?
- A line segment with endpoints at #(5, 5)# and #(1, 2)# is rotated clockwise by #pi/2#. What are the new endpoints of the line segment?

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