Circle A has a center at #(2 ,5 )# and a radius of #2 #. Circle B has a center at #(4 ,2 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(2  ,5   )# and a radius of #2    #.  Circle B has a center at #(4  ,2   )# and a radius of #3 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Allison Allen · Accepted Answer

Yes, the circles do overlap. See explanation.

Circle A, center(2,3), r=2.
Circle B, center(4,2), r=3

In the graph above, the red one is Circle A, and the green one Circle B. The white line is the distance between the two center points.

Distance between the centers of the two circles =
#sqrt((4-2)^2+(2-5)^2) = sqrt(2^2+3^2) = sqrt13 = 3.61#

As the distance between the two centers (3.61) is smaller than the sum of the two radii (2+3=5), the two circles overlap.

Theodore Abad · Answer

undefined To determine if the circles overlap, we need to 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, the circles do not overlap.

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Let's calculate the distance between the centers of Circle A and Circle B:

Center of Circle A: $(2, 5)$

Center of Circle B: $(4, 2)$

Distance $d = \sqrt{(4 - 2)^2 + (2 - 5)^2}$

$d = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}$

Now, compare the distance $d$ to the sum of the radii of the circles:

Radius of Circle A: 2

Radius of Circle B: 3

Sum of radii: $2 + 3 = 5$

Since $\sqrt{13} > 5$, the circles do not overlap.

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

Smallest distance = $ \sqrt{13} - 5 $.