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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 .

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  • $\begingroup$ what are $B,C$? $\endgroup$ – Anurag A Jan 8 '17 at 9:15
  • $\begingroup$ Perhaps $ABC$ is an equilateral triangle, and $O$ is inside $ABC$ such that $BO = 3$, $CO = 4$ and $AO = 5$. $\endgroup$ – JimmyK4542 Jan 8 '17 at 9:28
  • $\begingroup$ Perhaps...and perhaps not. Let's see if the OP explains his own question and, also, adds some work he may have done on it. $\endgroup$ – DonAntonio Jan 8 '17 at 9:52
  • $\begingroup$ Yes...Its that....JimmyK4542 $\endgroup$ – DEBOJJAL Jan 8 '17 at 12:48
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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.

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