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# geometry

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M is the midpoint of PQ and N is the midpoint of PR, and O is the intersection of QN and RM, as shown. If QN is perpendicular to PR, QN = 18, and PR = 14, then find OR.

Jul 27, 2021

Using mass points, let $m_P=1$, then $m_R=1$, $m_Q=1$, and we find $m_N=2$. Thus $ON=\frac{18}{3}=6$. Now, by the Pythagorean theorem, $OR=\sqrt{6^2+7^2}=\sqrt{85}$.