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Triangle Inside a Parallelogram

Triangle KNL is constructed so that it shares one of its sides (KL) with one of the sides of parallelogram HKLI. Also, the vertex N of triangle KNL, not on the shared side, can be anywhere on the opposite side (HI) of the parallelogram. What is the probability that a random point inside the parallelogram will also be inside the triangle?

InterMath

Re: Triangle Inside a Parallelogram

Hey jthomp18.

If the point is purely random then it will depend only on the ratio of the area between the two objects and the reason relates to a random process being uniform.

A purely random process has maximum entropy and a process with maximum entropy is uniform, although in this case entropy doesn't make a lot of sense since entropy in its proper form makes sense for discrete event spaces and not continuous (even though there is a continuous analog).

This translates into a uniform PDF that has the same probability across the entire parallelogram with density function P(X = x, Y = y) = 1/Area_Of_Parallelogram.

So now you have to consider P(Point Inside Triangle|Point Inside Quad) but Triangle is a subset of Quad so P(Point Inside Triangle AND Point Inside Quad) = P(Point Inside Triangle) so P(Point Inside Triangle|Point Inside Quad) = P(Point Inside Triangle)/P(Point Inside Quad) which is why its a ratio of the areas axiomatically.