Circle A has a center at #(1 ,-4 )# and a radius of #2 #. Circle B has a center at #(9 ,3 )# and a radius of #5 #. Do the circles overlap? If not what is the smallest distance between them?
The circles do not overlap;
they are separated by a minimum distance of
Circle A covers 2 (units) of the distance along the line segment from the center of A to the center of B.
Circle B covers 5 (units) of the distance along the line segment form the center of B to the center of A.
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The circles do not overlap. The distance between their centers can be calculated using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the given coordinates: Distance = √((9 - 1)^2 + (3 - (-4))^2) = √((8)^2 + (7)^2) = √(64 + 49) = √113 ≈ 10.63
The sum of the radii of the two circles is 2 + 5 = 7. Since the distance between the centers of the circles (10.63) is greater than the sum of their radii (7), the circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii:
Smallest distance = Distance between centers - Sum of radii = 10.63 - 7 = 3.63
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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.
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 #(1 ,4 )#, #(7 ,5 )#, and #(3 ,2 )#. How far is the triangle's centroid from the origin?
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- Circle A has a center at #(-4 ,-1 )# and a radius of #3 #. Circle B has a center at #(1 ,3 )# and a radius of #2 #. Do the circles overlap? If not what is the smallest distance between them?
- A line passes through #(5 ,0 )# and #(7 ,3 )#. A second line passes through #(3 ,1 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A line passes through #(1 ,5 )# and #(8 ,7 )#. A second line passes through #(3 ,6 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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