In the below diagram, various segment lengths in rectangle ABCD have been marked and segment EG bisects angle DEF. What is the length of segment EG?

EF is 17 by pythagorean theorem

angle FEC is arccos(8/17)
angle GED is (180 - arccos(8/17))/2 = 90 - arccos(8/17)/2

construct a perpendicular to CD through G.  Call the intersection H.
GH = 20
sin(90-arccos(8/17)/2)) = 20/x

x = 20/sin(90-arccos(8/17)/2))
=20/cos(arccos(8/17)/2))
by the half angle identity
=20/sqrt((cos(arccos(8/17))+1)/2
=20/sqrt(25/34)
=20*sqrt(34)/sqrt(25)
=4*sqrt(34) ≈23.32380758

Posted by Jer on 2010-11-08 14:57:42