All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
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.

See The Solution Submitted by Dennis    
Rating: 3.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Hints/Tips AE = 2BD implies B = 90 degrees | Comment 4 of 5 |
OOPS - Should not of put the hint in the title.

The length of the medians are
                      a
   |AE|^2 = bc[1 - (-----)^2]
                     b+c
        and
                      b
   |BD|^2 = ac[1 - (-----)^2]
                     a+c
Therefore, |AE| = 2|BD| implies
   b(b+c-a)(a+c)^2 = 4a(a+c-b)(b+c)^2    (1)
From the Law of Cosines,
   c^2 = a^2+b^2-2ab*cos(C)
       or
   c^2 = a^2+b^2-ab                      (2)
Equations (1) and (2) imply (with a lot of algebra)
   (4a^2-b^2)[c(a+b)+(a^2+b^2)] = 0
This implies
   b = 2a
this with equation (2) gives 
   c = 3*sqrt(3)
Therefore,
   b^2 = (2a)^2 = a^2+(a*sqrt(3))^2 
       = a^2+c^2
Therefore,
   B = 90 degrees
 

Edited on February 12, 2008, 4:13 pm

Edited on February 12, 2008, 4:15 pm

Edited on February 12, 2008, 11:53 pm
  Posted by Bractals on 2008-02-12 16:11:40

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (7)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information