In right triangle $ABC$, we have $AB = 10$, $BC = 24$, and $\angle ABC = 90^\circ$. If $M$ is on $\overline{AC}$ such that $\overline{BM}$ is a median of $\triangle ABC$, then what is $\cos \angle ABM$?

michaelcai
Nov 15, 2017

#1**+2 **

cos(ABM) = ?

\(AM=MC=BM=r(\text{circle })\)

\(\tan{ACB}=\frac{10}{24} \)

\(\frac{\sin(ACB)}{r}=\frac{\sin(90^{\circ}-ABM)}{r}\\ \sin(ACB)=\cos(ABM)\\ \cos(ABM)=\sin(\arctan(\frac{10}{24}))=0.384615384615...\\ ABM=\arccos(\frac{5}{13})=67.380135051983^{\circ } \)

heureka

Guest Nov 15, 2017

edited by
Guest
Nov 15, 2017