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Angle Bisectors (Posted on 2008-02-11) Difficulty: 3 of 5
In triangle ABC, angle C has a measure of 60 degrees. Point D lies on side AC so that BD bisects angle B and BD = 1. Similarly, point E lies on side BC so that AE bisects angle A and AE = 2.

Find the area of triangle ABC.

  Submitted by Dennis    
Rating: 3.0000 (2 votes)
Solution: (Hide)
Let x = the measure of angle CAE, y = the measure of angle CBD, 120 degrees - x = the measure of angle CEA, and 120 degrees - y = the measure of angle CDB.

AC/sin(120-x) = 2/sin(60) --> AC = (2/sqrt(3))(sqrt(3)cosx+sinx)

BC/sin(120-y) = 1/sin(60) --> BC = (1/sqrt(3))(sqrt(3)cosy+siny)


Now 2x+2y=120 degrees --> y=60-x -->

cosy=.5cosx+(sqrt(3)/2)sinx and siny=(sqrt(3)/2)cosx-.5sinx


So BC = (1/sqrt(3))(sqrt(3)cosx+sinx) --> AC = 2BC


Letting BC=v and AC=2v --> AB^2 = v^2 + 4v^2 - 4v^2cos60 -->


AB = v*sqrt(3) --> B is a right angle --> x=15 degrees.


Area = v^2*sqrt(3)/2 = (sqrt(3)/6)(sqrt(3)cosx+sinx)^2

Area = (sqrt(3)/6)(2+cos2x+sqrt(3)sin2x) = 1/2 + sqrt(3)/3

Comments: ( You must be logged in to post comments.)
  Subject Author Date
AnswerK Sengupta2009-01-07 15:29:17
Hints/TipsAE = 2BD implies B = 90 degreesBractals2008-02-12 16:11:40
re(2): SolutionBractals2008-02-11 20:47:24
Questionre: SolutionDennis2008-02-11 14:34:38
SolutionSolutionBractals2008-02-11 13:41:41
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