# Circle A has a center at #(8 ,3 )# and a radius of #1 #. Circle B has a center at #(4 ,4 )# and a radius of #2 #. Do the circles overlap? If not what is the smallest distance between them?

no overlap, ≈ 1.123

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

the 2 points here are (8 ,3) and (4 ,4) the centres of the circles.

sum of radii = radius of A + radius of B = 1 + 2 = 3

Since sum of radii < d , then no overlap

smallest distance = d - sum of radii = 4.123 - 3 = 1.123 graph{(y^2-6y+x^2-16x+72)(y^2-8y+x^2-8x+28)=0 [-10, 10, -5, 5]}

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To determine if the circles overlap, calculate the distance between their centers and compare it to the sum of their radii.

The distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)) is given by the formula:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Given:

- Center of circle A: ((8, 3))
- Center of circle B: ((4, 4))
- Radius of circle A: (r_A = 1)
- Radius of circle B: (r_B = 2)

Calculate the distance between the centers:

[ d = \sqrt{(4 - 8)^2 + (4 - 3)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123 ]

Now, compare this distance with the sum of the radii:

[ r_A + r_B = 1 + 2 = 3 ]

Since ( \sqrt{17} \approx 4.123 > 3 ), the circles do not overlap.

The smallest distance between them is the difference between the distance of their centers and the sum of their radii:

[ \text{Smallest distance} = \sqrt{17} - 3 \approx 4.123 - 3 = 1.123 ]

So, the smallest distance between the circles is approximately (1.123).

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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.

- Circle A has a center at #(3 ,2 )# and an area of #13 pi#. Circle B has a center at #(9 ,6 )# and an area of #28 pi#. Do the circles overlap?
- Circle A has a center at #(7 ,-2 )# and a radius of #2 #. Circle B has a center at #(4 ,2 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(11 ,5 )# and an area of #100 pi#. Circle B has a center at #(4 ,9 )# and an area of #36 pi#. Do the circles overlap? If not, what is the shortest distance between them?
- A triangle has corners at #(9 ,3 )#, #(7 ,4 )#, and #(3 ,1 )#. How far is the triangle's centroid from the origin?
- A line passes through #(2 ,8 )# and #( 1, 5 )#. A second line passes through #( 6, 1 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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