+36 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 side PR . If PR =17, PQ= 9, and QR=10, compute the length of XY.
+834 Let's break down the problem and use the properties of angle bisectors and similar triangles to solve it.
1. Angle Bisector Theorem:
The angle bisector of angle P divides side QR into segments proportional to the lengths of the adjacent sides PQ and PR.
Let's denote the lengths of the segments as follows:
QX = a
XR = b
So, we have the proportion:
a/b = PQ/PR = 9/17
2. Similar Triangles:
Triangle PXY and triangle PRX are similar because they share angle P, and angle PXY = angle PRX (both are right angles).
From the similarity of these triangles, we can set up the following proportion:
XY/PR = QX/PQ
Substituting the known values and the expression for QX from step 1:
XY/17 = (9a)/(9(9+17))
Simplifying:
XY = (17a)/(26)
3. Using the Angle Bisector Theorem Again:
From the proportion in step 1, we can write:
a + b = QR = 10
Substituting b = 10 - a into the proportion:
a/(10-a) = 9/17
Solving for a:
17a = 90 - 9a
26a = 90
a = 90/26
4. Finding XY:
Substituting the value of a into the expression for XY from step 2:
XY = (17 * (90/26)) / 26
Simplifying:
XY = 1530 / 676
Reducing the fraction:
XY = 45/26
Therefore, the length of XY is 45/26.
