Circle A has a center at #(5 ,4 )# and an area of #15 pi#. Circle B has a center at #(2 ,1 )# and an area of #90 pi#. Do the circles overlap?

Question

Circle A has a center at #(5   ,4  )# and an area of #15 pi#.  Circle B has a center at #(2  ,1   )# and an area of #90 pi#.  Do the circles overlap?

Wyatt Anderson · Accepted Answer

#"one circle inside other"# What we have to do here is #color(blue)"compare "# the distance (d) between the centres of the circles to the #color(blue)"sum/difference"# of the radii. #• " if sum of radii ">d" then circles overlap"# #• " if sum of radii"< d" then no overlap"# #• " if difference of radii">d" then circle inside other"# #• " area of circle "=pir^2larr" r is the radius"# #color(blue)"Circle A"# #rArrpir^2=15pirArrr=sqrt15~~3.873# #color(blue)"Circle B"# #rArrpir^2=90pirArrr=sqrt90~~ 9.486# #"to calculate d use the "color(blue)"gradient formula"# #color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))# #"let "(x_1,y_1)=(2,1)" and "(x_2,y_2)=(5,4)# #d=sqrt((5-2)^2+(4-1)^2)=sqrt18~~ 4.243# #"sum of radii "=3.873+9.486=13.359# #"difference of radii "=9.486-3.873=5.613# #"since difference of radii">d" then circle inside other"# graph{((x-5)^2+(y-4)^2-15)((x-2)^2+(y-1)^2-90)=0 [-40, 40, -20, 20]}

Allison Allen · Answer

undefined To determine if the circles overlap, calculate the distance between their centers. If the distance is less than the sum of their radii, they overlap.

The distance between the centers of Circle A and Circle B is √((5-2)^2 + (4-1)^2) = √(3^2 + 3^2) = √(18) = 3√2.

The radii of Circle A and Circle B are √(15π/π) = √15 and √(90π/π) = √90 = 3√10, respectively.

The sum of their radii is √15 + 3√10.

Since the distance between the centers (3√2) is less than the sum of their radii (√15 + 3√10), the circles overlap.