Problem:
In △ABC shown in the figure, AB=7,BC=8,CA=9, and AH is an altitude. Points D and E lie on sides AC and AB, respectively, so that BD and CE are angle bisectors, intersecting AH at Q and P, respectively. What is PQ?
Answer Choices:
A. 1
B. 853
C. 542
D. 1585
E. 56 Solution:
Let x=BH. Then CH=8−x and AH2=72−x2=92−(8−x)2, so x=2 and AH=45. By the Angle Bisector Theorem in △ACH,PHAP=CHCA=69, so AP=53AH. Similarly, by the Angle Bisector Theorem in △ABH,QHAQ=BHBA=27, so AQ=97AH. Then PQ=AQ−AP=(97−53)AH=45845=1585.
