Probability in Algebra 2 honors

Question

Hannah is playing a virtual reality game in which she must toss a disc to land on the largest triangular section on the board. If the disc is equally likely to land anywhere on the board, what is the probability that she will succeed?

Answer 1

I assume the board is rectangular.

Suppose A, B, C are points on the board (or the edges of the board) such that triangle ABC has the greatest area among such triangles.

I claim that A, B, and C must all lie on the boundary of the board (on a corner or on an edge).

Suppose one of the vertices, w.l.o.g. A, is not on the boundary of the board. Then there is a disc D centered at A that is entirely in the interior of the board. The area of ABC is half of the length of line segment BC times the length of a perpendicular dropped from A to the line BC. Follow that line segment away from A in the direction away from line segment BC, but staying in D, to a point A' in D. A'BC has base BC and altitude to A' greater than to A, contradicting the maximal area of ABC

If a vertex (again w.l.o.g. A) is not at a corner of the board, then it is on an edge parallel BC, the opposite edge of triangle ABC, because otherwise, moving A along the edge in one direction would decrease area ABC and moving the other direction would increase it, the latter being a contradiction to ABC's maximum area. So BC is parallel to a side of the board that A is on, so pushing it to coincide with that opposite side would increase ABC's area, BC must already be that opposite side. This triangle has area equal to half of the area of the board (with a side length as base, and an altitude parallel and equal in length to the other sides).

On the other hand, if all three of A, B, C are corners of the rectangle, then that triangle also has area half the area of the board.

So the probability of the disc landing in this triangle is approximately 0.5, exactly if the disc's location is given by its center, which can land anywhere on the board.