Week 13

Example 2. Triangle ABC is a right triangle with ∠ACB as its right angle, ∠ABC = 60° and AB = 10. Point P is randomly chosen inside ABC, and BP is extended to meet AC at D. What is the probability that BD > 5√2?

Of course, you can find the solution just below, but it is highly recommended that, you first try to solve it on your own.

Just remember the words of Paul Halmos, who says “the only way to learn mathematics is to do mathematics”.

Solution 1. From the given relation, we find that

Since (b + c) and (d + a) are ≠ 0 (otherwise, the given relation becomes undefined), only possibilities are a = c or a + b + c + d = 0.

Solution 2. Since AB = 10 and ∠ABC = 60°, we have

It is common in inequality problems to first consider the boundary case. Suppose that E is the point on AC with BE = 5√2. Then

and ∆ECB is an isosceles right triangle.

So BD < 5√2 when D is between E and C, that is, when P is inside ∆BEC. The probability that this will occur is

Hence the probability that BD > 5√2 is