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In triangle $PQR$, let $M$ be the midpoint of $\overline{PQ}$, let $N$ be the midpoint of $\overline{PR}$, and let $O$ be the intersection of $\overline{QN}$ and $\overline{RM}$, as shown. If $\overline{QN}\perp\overline{PR}$, $QN = 1$, and $PR =1$, then find $OR$.

 Feb 23, 2024
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O is the centroid of the triangle and divides the medians in ratio 2:1. So:
\(ON+QO=QN=1\\ \frac{ON}{QO}=\frac{1}{2}\\ ON=\frac{1}{3}\)
Also, N is the midpoint of \(PR\) so:
\(NR=\frac{PR}{2}=\frac{1}{2}\)
 

Finally using pythagorean theorem on \(\triangle ONR\) we get:
\(OR=\sqrt{NR^2+ON^2}\\ OR=\sqrt{\frac{1}{3}^2+\frac{1}{2}^2}\\ \boxed{OR=\frac{\sqrt{13}}{6}}\)

 Feb 24, 2024

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