Right angled median intersection question

Question

All the information is included in the image. Find the length AB.

Only clue I have is that the length CX is 2.5 where X is the perpendicular foot of B which i found out geometrically. However I don't know how to find CX mathematically.

Answer 1

It is a well known problem that says that if two medians are perpendicular then $ a^2+b^2=5 c^2 $ or something like that. The proof uses the formula for the length of the median, the fact that medians cut each other in a 2:1 ratio, the Pythagorean theorem then some expansion etc.

Answer 2

Using Pythagoras' theorem and basic facts about medians will be enough: set $AA'=3a$, $BB'=3b$ and let $G$ be the inersection of $(AA')$ and $(BB')$. As $AG=2a$, $BG=2b$, Pythagoras in the right triangles $AGB', A'GB$ and $AGB$ says that $$4a^2+b^2=4, \quad a^2+4b^2=\frac 94, \quad AB^2=4(a^2+b^2)$$ Adding the first two equations we have $\,5(a^2+b^2)=\dfrac{25}4 $, whence $$AB^2=5\enspace\text{or}\enspace AB=\sqrt 5.$$