Circle A has a center at #(5 ,4 )# and a radius of #4 #. Circle B has a center at #(6 ,-8 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(5  ,4   )# and a radius of #4    #.  Circle B has a center at #(6   ,-8   )# and a radius of #2 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Theodore Abad · Accepted Answer

The circles do not overlap.
Smallest distance#=d-S=12.04159-6=6.04159 " "#units

From the given data:
Circle A has a center at (5,4) and a radius of 4. Circle B has a center at (6,−8) and a radius of 2. Do the circles overlap? If not, what is the smallest distance between them?

Compute the sum of the radius:
Sum #S=r_a+r_b=4+2=6" "#units

Compute the distance from center of circle A to center of circle B:

#d=sqrt((x_a-x_b)^2+(y_a-y_b)^2)#

#d=sqrt((5-6)^2+(4--8)^2)#

#d=sqrt((-1)^2+(12)^2)#

#d=sqrt145=12.04159#

Smallest distance#=d-S=12.04159-6=6.04159#

Sadie Adams · Answer

undefined To determine if the circles overlap, we can calculate the distance between their centers and compare it to the sum of their radii. If the distance between the centers is less than the sum of the radii, the circles overlap. Otherwise, they do not overlap.

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

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

For Circle A with center $(5, 4)$ and Circle B with center $(6, -8)$, the distance between their centers is:

$$ \sqrt{(6 - 5)^2 + (-8 - 4)^2} = \sqrt{1^2 + (-12)^2} = \sqrt{1 + 144} = \sqrt{145} $$

The sum of the radii of Circle A and Circle B is $4 + 2 = 6$.

Since the distance between the centers of the circles ($\sqrt{145}$) is greater than the sum of their radii (6), the circles do not overlap.

To find the smallest distance between the circles, we subtract the sum of their radii from the distance between their centers:

$$ \sqrt{145} - 6 $$