Equilateral Triangle Question

Question

In a equilateral triangle ABC , $3$ Rods of length $3 , 4 , 5$ units are placed such that they intersect at a common point $O$ and other end being on the vertex A,B,C respetively.. Find the angle $BOC$, if $BO=3$ units and $CO=4$ units .

Answer 1

HINT.

Use the cosine rule to obtain two equations for $x=\cos(\angle BOC)$ and $y=\cos(\angle AOC)$. Remember that $$ \cos(\angle AOB)=\cos(2\pi-\angle AOC-\angle BOC)=\cos(\angle AOC+\angle BOC)=xy-\sqrt{1-x^2}\sqrt{1-y^2}. $$ You'll get in the end a third degree equation for $x$, which has a rational solution easy to find and can thus be factorized.