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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 8, 2023
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We can solve this problem using the angle bisector theorem and some basic geometry:

First, since PQR is an isosceles triangle with PR = QR, we know that the angle bisector of angle P also bisects QR, so X is the midpoint of QR.

Next, since XY is perpendicular to PR, we can use the Pythagorean theorem to find its length. Let's call the length of XY h. Then, using the fact that QX = RX = QR/2 = 4.5, we have:

h^2 = PX^2 - QX^2          (by the Pythagorean theorem)
     = (PQ*QR/PQ + QR)*QR/4 - QR^2/4  (by the angle bisector theorem)
     = 81/4 - 81/4
     = 0

Therefore, h = 0, which means that XY coincides with PR. So, the length of XY is equal to the length of PR, which is 9. 

 May 8, 2023

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