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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.

 Jun 26, 2023
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Let's write d to represent XY's length. Trigonometry can be used to determine the length of XY since triangle PXY is a right triangle. Specifically, we have

tan(30) = XY / PY

Since PY = PQ - QY and PQ = QR = 9, we have PY = 9 - QY. Therefore:

tan(30) = XY / (9 - QY)

Solving for XY, we get:

XY = (9 - QY) tan(30)

QY needs to be located. The Pythagorean theorem can be used to determine QY because triangle QYX is also a right triangle:

QY² + XY² = QX²

Triangle QRX being a 30-60-90 triangle gives us:

QX = QR / 2 = 4.5

Therefore:

QY² + XY² = 4.5²

QY² + (9 - QY)² tan²(30) = 4.5²

The result of simplifying and solving for QY is:

QY = 2.25

Inputting this value into the XY expression yields the following result:

XY = (9 - 2.25) tan(30) = 2.25 × √(3)

 Jun 27, 2023

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