+556 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 Q to side PR. If PQ = 3, PR = 4, and QR = 5, and then compute the length of XY.
+129830 Q
5
3 X
P R
4
Y = P
Angle QPR = 90°
sin angle (QRP) = (3/5)
Angle RPX = 45°
sin angle RPX = 1/sqrt 2
Because PX is an angle bisector
QP / QX = PR / RX
Let RX = x QX = 5 - x
So
3/ (5-x) = 4 / x
3x = 4 (5-x)
3x = 20 - 4x
7x = 20
x = 20/7
Law of Sines
(YX) /sin (QRP) = RX / sin (RPX)
YX / (3/5) = (20/7) / ( 1/sqrt 2)
YX = (3/5)(20/7) * sqrt 2 = (60/35)sqrt 2 = (12/7)sqrt (2) ≈ 2.42
