# Circle A has a center at #(2 ,5 )# and a radius of #2 #. Circle B has a center at #(4 ,-1 )# and a radius of #6 #. Do the circles overlap? If not, what is the smallest distance between them?

Yes

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The circles do not overlap. The distance between their centers is ( \sqrt{(4-2)^2 + (-1-5)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} ) units. The sum of their radii is ( 2 + 6 = 8 ) units. Since the distance between their centers 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, which is ( 6\sqrt{2} - 8 ) 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.

- What is the perimeter of a triangle with corners at #(1 ,5 )#, #(6 , 2 )#, and #(5 ,7 )#?
- Express the area of a triangle given by vertices, #A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)#. Show that it can be expressed as determinant of: #det(Delta) = [(1, 1, 1 ),(x_1, x_2, x_3),(y_1, y_2, y_3) ]#.Calculate the area of A(3,6), B(7,8), & C(5,2)?
- Circle A has a center at #(1 ,7 )# and a radius of #3 #. Circle B has a center at #(-3 ,-2 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?
- A line passes through #(4 ,7 )# and #(2 ,4 )#. 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 #(1 ,4 )# and an area of #100 pi#. Circle B has a center at #(7 ,9 )# and an area of #36 pi#. Do the circles overlap? If not, what is the shortest distance between them?

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