Circle A has a center at #(2 ,3 )# and a radius of #5 #. Circle B has a center at #(3 ,8 )# and a radius of #1 #. Do the circles overlap? If not what is the smallest distance between them?

Question

Circle A has a center at #(2   ,3  )# and a radius of #5 #.  Circle B has a center at #(3   ,8  )# and a radius of #1 #.  Do the circles overlap?  If not what is the smallest distance between them?

Sadie Allen · Accepted Answer

If the two circles touched at a single point , then the distance between the centers would be #5+1=6# and the distance between the circles would be zero.

Using the distance formula, the actual distance between the two centers is:

#sqrt[(8-3)^2+(3-2)^2]=sqrt26~~5.1#

Since the distance #5.1<6#, the two circles must overlap !

Hope that helped

Sophia Alfred · Answer

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

The distance between the centers of Circle A and Circle B can be calculated using the distance formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

For Circle A with center $(2, 3)$ and Circle B with center $(3, 8)$:
$Distance = \sqrt{(3 - 2)^2 + (8 - 3)^2}$
$Distance = \sqrt{1^2 + 5^2}$
$Distance = \sqrt{1 + 25}$
$Distance = \sqrt{26}$
$Distance \approx 5.099$.

The sum of the radii of Circle A and Circle B is $5 + 1 = 6$.

Since the distance between the centers (approximately 5.099) is greater than the sum of the radii (6), the circles do not overlap.

The smallest distance between the circles is the difference between the distance between their centers and the sum of their radii: $5.099 - 6 = -0.901$. However, since distances cannot be negative, the smallest distance between the circles is 0.