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

Question

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

Caroline Aucoin · Accepted Answer

There is no overlap.

We can find the coordinates of the endpoints of the horizontal and vertical diameters of the circles by adding and subtracting the radius from the coordinates of the circles' centres.

Circle A has a horizontal diameter that runs from #(0,-2)# to #(4,-2)# and a vertical diameter that runs from #(2,-4)# to #(2,0)#.

Circle A therefore has the domain #{0,4}# and range #{-4,0}#

Circle B has a horizontal diameter that runs from #(-3,-4)# to #(5,-4)# and a vertical diameter that runs from #(1,-8)# to #(1,0)#

Circle B has domain #{-3,5}# and range #{-8,0}#

The domain of circle A is completely within the domain of circle B, and likewise the range of circle A is completely within the range of circle B, so circle A falls completely within circle B.

Wyatt Anderson · Answer

undefined Yes, the circles overlap. The smallest distance between them is 1 unit.

Autumn Armstrong · Answer

undefined To determine if the circles overlap, we need to compare the distance between their centers to the sum of their radii. If the distance between the centers is greater than the sum of their radii, then the circles do not overlap.

The distance between two points (x1, y1) and (x2, y2) is given by the formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)

For circle A, the center is at (2, -2), and for circle B, the center is at (1, -4).

Distance between the centers:
distance = √((1 - 2)^2 + (-4 - (-2))^2)
          = √(1 + 4)
          = √5

Sum of the radii:
radius of circle A = 2
radius of circle B = 4
sum of radii = 2 + 4 = 6

Since the distance between the centers (√5) is less than the sum of their radii (6), the circles do overlap.

If the circles did not overlap, we could find the smallest distance between them by subtracting the sum of their radii from the distance between their centers.

But in this case, since the circles overlap, there's no need to calculate the smallest distance between them.