Circle A has a radius of #2 # and a center at #(8 ,3 )#. Circle B has a radius of #3 # and a center at #(4 ,5 )#. If circle B is translated by #<-3 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
There's no overlap here. The closest points will be
Circle B's translated center is
If the circles are to overlap, the distance between their centers must be less than or equal to the sum of the radii,
So there's no overlap here. The closest points will be
This was pretty easy, but sometimes when we're doing these sorts of problems we end up asking if the sum of two square roots is more or less than a third square root. Of course those can be done with a calculator, but there is a way to compare the squared lengths directly. It uses Archimedes' Theorem for the area
A real triangle, one whose sides satisfy the triangle inequality, is one with a real area. So we require
or, when the degenerate triangle formed from three collinear points is acceptable,
This is all three constraints of the triangle inequality in a single inequality. We don't have to check
In our example,
Nope, not a real or degenerate triangle, so no overlap.