How to Find the Longest Side of Triangle ABC with Given Medians?

[Ilikebugs]

In triangle ABC, ∠ABC = 90◦
. A median is drawn from A meeting BC at M such
that AM = 5. A second median is drawn from C meeting AB at N such that
CN = 2√10.
Determine the length of the longest side of triangle ABC

I have no idea where to even start on this one

 

[Euge]

Welcome, Ilikebugs! (Wave)

Let $BM = x$ and $NB = y$. Since $M$ is the median of $BC$, $BC = 2 BM = 2x$; since $N$ is the median of $AB$, $AB = 2NB = 2y$. Now try to do the following:

1. Find an expression for $AC$ in terms of $x$ and $y$ (Hint: Use the Pythagorean Theorem).
2. Consider triangle ABM, and write an equation involving $x$ and $y$ using the Pythagorean theorem. Do the same for triangle $NBC$.
3. Solve the resulting system of equations. Take your $x$-solution and $y$-solution, and plug it into the expression for $AC$ to obtain the answer.