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  4. Consider a triangle P Q R with coordinates of its vertices as P(-8,5), Q(-15,-19), and R(1,-7). The bisector of the interior angle of P has the equation which can be written in the form ax +2 y+c=0. The distance between the orthocenter and the circumcenter of triangle P Q R is

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Q. Consider a triangle $P Q R$ with coordinates of its vertices as $P(-8,5), Q(-15,-19)$, and $R(1,-7)$. The bisector of the interior angle of $P$ has the equation which can be written in the form ax $+2 y+c=0$. The distance between the orthocenter and the circumcenter of triangle $P Q R$ is

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Solution:

Since the triangle is right-angled, the circumcenter is the midpoint of $P Q$ and the orthocenter is $R(1,-7)$. Hence,
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$R M=\left|\sqrt{\left(\frac{23}{2}+1\right)^{2}}\right|=12 \frac{1}{2}$