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?

Question

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?

Mason Adams · Accepted Answer

circles overlap

First step is to find the distance between the centres using the #color(blue)" distance formula "# # d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) # where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points " # let # (x_1,y_1)=(11,5)" and "(x_2,y_2)=(4,9)# substitute these values into the formula to find d # d = sqrt((4-11)^2 + (9-5)^2) = sqrt(49+16) = sqrt65 ≈ 8.06# Now, require to find the radii of circles A and B, given that the areas are known. Circle A : # pir^2 = 100pi rArr r^2 = (100pi)/pi = 100 rArr r = 10 # Circle B: #pir^2 = 36pi rArr r^2 = 36 rArr r = 6 # radius of circle A + radius of circle B = 10 + 6 = 16 since sum of radii > distance between centres → overlap

Violet Aldrich · Answer

undefined The circles do not overlap. The shortest distance between them can be found by calculating the distance between the centers of the circles and subtracting the sum of their radii. For Circle A with center (11, 5) and Circle B with center (4, 9), the distance between their centers is approximately 7.8102 units. The sum of their radii is approximately 5.64 units. Therefore, the shortest distance between the circles is approximately 2.1702 units.