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In triangle PQR, let X be the intersection of the angle bisector of angle P with side QR, and let Y be the foot of the perpendicular from X to line PR.  If PQ = 9, QR = 9, and PR = 9, then compute the length of XY.

 May 17, 2023
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Since triangle \(ABC\) is equilateral, \(PX\) is also a median, bisecting side \(QR\). Thus, \(XR=\frac{9}{2}.\) Note that triangle \(YXR\) is a 30-60-90 triangle with right angle at \(Y\) and 30 degree angle at \(\angle{YXR}\). Thus, \(YR = \frac{\frac{9}{2}}{2} = \frac{9}{4}.\) Since the side opposite the \(60^\circ\) angle in a 30-60-90 triangle is \(\sqrt{3}\) times the side opposite the \(30^\circ\) angle, we have \(XY = \frac{9}{4} \cdot \sqrt{3} = \boxed{\frac{9\sqrt{3}}{4}}.\)

 May 17, 2023

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