# Circle A has a radius of #2 # and a center of #(7 ,6 )#. Circle B has a radius of #3 # and a center of #(5 ,3 )#. 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?

circles overlap

What we have to do here is compare the distance (d) between the centres with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

The first step is to find the new centre of B under the given translation. Under a translation the shape of the figure does not change only it's position.

centre of B (5 ,3) → (5-1 ,3+2) → (4 ,5)

The 2 points here are (7 ,6) and (4 ,5)

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

Since sum of radii > d , then circles overlap. graph{(y^2-12y+x^2-14x+81)(y^2-10y+x^2-8x+32)=0 [-20, 20, -10, 10]}

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To determine if circle B overlaps circle A after being translated, calculate the distance between the centers of the circles after the translation. If this distance is less than the sum of the radii of both circles, then the circles overlap. Otherwise, they do not overlap. If the circles do not overlap, the minimum distance between points on both circles is equal to the difference between the distance between the centers and the sum of their radii.

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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 #(3, 2 )#, ( 5, 7)#, and #( 2, -5)#. If the triangle is reflected across the x-axis, what will its new centroid be?
- A line segment has endpoints at #(1 ,4 )# and #(3 ,5 )#. The line segment is dilated by a factor of #6 # around #(2 ,5 )#. What are the new endpoints and length of the line segment?
- Circle A has a radius of #4 # and a center of #(7 ,3 )#. Circle B has a radius of #2 # and a center of #(1 ,2 )#. 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?
- A triangle has corners at #(0, 5 )#, ( 1, -6)#, and #(8, -4 )#. If the triangle is reflected across the x-axis, what will its new centroid be?
- A line segment has endpoints at #(6 ,4 )# and #(2 ,5 )#. If the line segment is rotated about the origin by # pi #, translated horizontally by # 2 #, and reflected about the x-axis, what will the line segment's new endpoints be?

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