Circle A has a center at #(2 ,2 )# and a radius of #5 #. Circle B has a center at #(12 ,8 )# and a radius of #1 #. Do the circles overlap? If not what is the smallest distance between them?
The circles do not overlap.
There is a minimum distance of
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To determine if the circles overlap, we can calculate the distance between their centers and compare it to the sum of their radii. The distance between two points ((x_1, y_1)) and ((x_2, y_2)) is given by the formula:
[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
For Circle A with center ((2, 2)) and Circle B with center ((12, 8)):
[ \text{Distance} = \sqrt{(12 - 2)^2 + (8 - 2)^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.66 ]
The sum of their radii is (5 + 1 = 6). Since the distance between their centers ((\sqrt{136})) is greater than the sum of their radii, the circles do not overlap.
The smallest distance between them is the difference between the distance between their centers and the sum of their radii:
[ \text{Smallest Distance} = \text{Distance} - (\text{Radius of Circle A} + \text{Radius of Circle B}) ]
[ \text{Smallest Distance} = \sqrt{136} - (5 + 1) = \sqrt{136} - 6 \approx 5.66 ]
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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.
- Circle A has a center at #(2 ,2 )# and an area of #18 pi#. Circle B has a center at #(13 ,6 )# and an area of #27 pi#. Do the circles overlap?
- What is the perimeter of a triangle with corners at #(7 ,5 )#, #(8 ,2 )#, and #(4 ,7 )#?
- Circle A has a center at #(1 ,5 )# and a radius of #3 #. Circle B has a center at #(2 ,-3 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?
- A triangle has corners at #(2 ,4 )#, #(8 ,6 )#, and #(4 ,9 )#. How far is the triangle's centroid from the origin?
- A triangle has corners at #(9 ,5 )#, #(2 ,7 )#, and #(3 ,4 )#. How far is the triangle's centroid from the origin?

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