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Length of Angle Bisector (Posted on 2010-11-07) Difficulty: 3 of 5
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?

See The Solution Submitted by Charlie    
Rating: 2.0000 (2 votes)

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Solution trig solution worked out Comment 3 of 3 |
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
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