Towns A and B are 16 miles apart. How many points are 10 miles from town A and 12 miles from town B?
This is essentially the intersection of 2 circles, so we can have either 0, 1, or 2 points of intersection. In this question we have 2.
Let's figure this out by imagining we are building fences around towns A and B.
The first thing to check is will a fence around town A (which I will call fence A) and around town B (which I will call fence B) will intersect each other.
Since we are building fences that are the same distance away all the way around the respective towns, we're building round fences.
So now there are 3 possible answers:
- if the towns are father apart than the distance each fence reaches, they won't touch,
- if the towns are exactly the distance apart that the two fences reach, then the fences will intersect in one place, and
- if the towns are closer together than the 2 fences reach, the fences will intersect in 2 places.
Since Fence A will reach 10 miles towards Town B and Fence B will reach 12 miles towards Town A, the 2 together reach 22 miles. But the towns are closer than 22 miles apart - they are 16 miles apart, and so the fences will intersect in 2 places.
(Imagine Town A is on the left and the circle around it is Fence A. Town B is on the right and the circle around it is Fence B.)
I should note (thanks to @George C for catching this) that the solution I gave assumes that the distance between the 2 towns is relatively large compared to the radii of the 2 fences. If the towns were relatively close compared to the radii of the fences, then we'd have the same 3 potential solutions of 0, 1, and 2 points of intersection, but the fences could be 1 completely inside the other (so no points), could touch at one point, or could intersect and have 2 points.