# Circle A has a center at #(-2 ,-4 )# and a radius of #2 #. Circle B has a center at #(-3 ,2 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?

The circles overlap

Along the line segment joining the two centers: circle A covers 2 units and circle B covers 5 units

for a total of 7 units covered by the circles.

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The circles do not overlap. The distance between their centers can be calculated using the distance formula, which is the square root of the sum of the squares of the differences in their coordinates. The distance between the centers of Circle A and Circle B is approximately ( 6.403 ) units.

The sum of their radii is ( 2 + 5 = 7 ) units. Since the distance between their centers is greater than the sum of their radii, the circles do not overlap.

The smallest distance between the circles is the difference between the distance between their centers and the sum of their radii. So, the smallest distance between the circles is approximately ( 6.403 - 7 = -0.597 ) units. However, distances are non-negative, so we consider the absolute value. Thus, the smallest distance between the circles is approximately ( 0.597 ) units.

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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 passes through #(9 ,3 )# and #( 4, 5 )#. A second line passes through #( 7, 8 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A line passes through #(2 ,8 )# and #(4 ,5 )#. A second line passes through #(3 ,5 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- Circle A has a center at #(9 ,4 )# and a radius of #3 #. Circle B has a center at #(-1 ,6 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?
- What is the midpoint of the segment whose endpoints are (-1, 4) and (3, 6)?
- A line passes through #(3 ,2 )# and #(7 ,3 )#. A second line passes through #(1 , 4 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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