Circle A has a center at #(8 ,5 )# and an area of #100 pi#. Circle B has a center at #(4 ,2 )# 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 #(8   ,5  )# and an area of #100 pi#.  Circle B has a center at #(4   ,2   )# and an area of #36 pi#.  Do the circles overlap?  If not, what is the shortest distance between them?

Sadie Adams · Accepted Answer

circles overlap

What we have to do here is compare the distance ( d) between the centres of the circles to the sum of the radii. • If sum of radii > d , then circles overlap • If sum of radii < d , then no overlap Before doing this we require to find the radii of the circles. #color(orange)"Reminder"# the area (A) of a circle is. #color(red)(|bar(ul(color(white)(a/a)color(black)(A=pir^2)color(white)(a/a)|)))# #color(blue)"Circle A " pir^2=100pirArrr^2=(100cancel(pi))/cancel(pi)rArrr=10# #color(blue)"Circle B "pir^2=36pirArrr^2=(36cancel(pi))/cancel(pi)rArrr=6# To calculate d, use the #color(blue)"distance formula"# #color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))# where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"# The 2 points here are (8 ,5) and (4 ,2) the centres of the circles. #d=sqrt((8-4)^2+(5-2)^2)=sqrt(16+9)=sqrt25=5# sum of radii = radius of A + radius of B = 10 + 6 = 16 Since sum of radii > d , then circles overlap graph{(y^2-4y+x^2-8x-16)(y^2-10y+x^2-16x-11)=0 [-9.86, 9.88, -4.93, 4.935]}

Caroline Aucoin · Answer

undefined To determine whether the circles overlap, we need to calculate the distance between their centers and compare it to the sum of their radii.

The formula to calculate the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a two-dimensional plane is given by the distance formula:

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

Given the centers of Circle A at $(8, 5)$ and Circle B at $(4, 2)$, we can plug these values into the distance formula to find the distance between their centers.

Next, we compare this distance to the sum of the radii of the two circles. If the distance between the centers is greater than the sum of the radii, then the circles do not overlap.

If the distance between the centers is less than the sum of the radii, then the circles overlap.

To find the shortest distance between the circles if they do not overlap, we subtract the sum of the radii from the distance between their centers.

Let's perform these calculations.