Question

Circle A has a center at `(5 ,-2 )` and a radius of `2`. Circle B has a center at `(4 ,6 )` and a radius of `3`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

They do not overlap; there is a minimum distance between them of `3.06` units (approximately)

Explanation:

Given circles with centers `(5,-2)` and `(4,6)`
the distance between the centers can be calculated using the Pythagorean Theorem:
`color(white)("XXX")d=sqrt((4-5)^2+(6-(-2))^2)=sqrt((-1)^2+8^2)=sqrt(65)`

Using a calculator `d~~8.06`

Measuring along a line joining the two centers:
one circle's radius takes up 2 units and
the other circle's radius takes up 3 units.

Since the sum of the two radii do not make up the distance between the centers, the circles do not overlap,
and the minimum distance between them is (approximately)
`color(white)("XXX")8.06-(2+3)=3.06`