Question

PLEASEEEE HELPPP!! 30 POINTS! PROVE THAT AM IS PERPENDICULAR TO BM

Answer 1

Extend the line segment AM until it touches AC, and call that intersection point P. Triangle AMP then forms a pair of similar triangles:

$\Delta AMP\sim\Delta PDM$

From this we know that angles MDP and AMP are congruent, and we already know MDP is a right angle so it follows that $AM\perp BM$.

(You can ignore most of the labels in the image; I realized moments later after posting that the proof is much simpler than I originally thought)