# Todai Entrance Exam: Math 2019 – Problem 3

**(1)**

**(2)**

The easy way:

The projection of on is . So .

The hard way:

Let be the intersection between and . As the triangle ABC is symmetric with respect to the line , every point on has a corresponding point on . So:

Note that as .

**(3)**

We need to find and . First, we find :

Now, we find . We can represent coordination in terms of .

Note that:

So:

Therefore:

Now compute :

Note that .

**(4)**

Let :

Let , and thanks to math.com:

**(5)** The same procedure can be applied to . Let be the expectation of given is located on , then: