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.
+131 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)
