+0  
 
0
38
4
avatar+241 

 

We have a triangle \(\triangle ABC \) such that \(AB = BC = 5\) and \(AC = 4. \)  If AD is an angle bisector such that D is on  BC then find the value of \(AD^2.\) Express your answer as a common fraction.

 
RektTheNoob  Jan 13, 2018
Sort: 

4+0 Answers

 #1
avatar+80978 
+2

Probably more elegant ways to do this.....but....

 

Here's a diagram :

 

 

 

 

AD  will bisect BAC.......and this angle bisector  creates the following ratio

 

BA / AC  =  BD / DC

5/4  =  BD / DC

 

Let  DC  =  x     then BD  = (5/4)x

And  DC + BD  = 5        so

x + (5/4)x  = 5

(9/4)x  =  5

x  = 20/9  =  DC

 

Draw  altitude  BG to AC .....this will bisect AC....   and  draw DF perpendicular to AC

Now......BG  =  √[BC^2 - GC^2]  = √[ 5^2 - 2^2]  = √21

 

And   triangles BGC  and DFC  are similar

 

And  BG/ BC   is similar to  DF/DC......so

√21 / 5  =  DF/ (20/9)     so

 

(√21 / 5) * (20/9)  =  DF   = 4√21 / 9  

 

Likewise

GC /  BC  =  FC /DC

2/5  =  FC / (20/9)

(20/9)(2/5)  = FC  =  40/45  = 8/9

 

And AF  =  AC  - FC  =   4  -  8/9     =  28/9

 

But  ADF  forms a right triangle  with   AF^2  + DF^2   =  AD^2

AF^2  =  (28/9)^2  =   28^2 / 81  =  784/81

DF^2  = [ 4√21 / 9 ]^2  =   16*21 / 81  =  336/81

 

 

So    AD^2  =    [ 336 + 784 ] 81  =  1120/81

 

 

cool cool cool

 
CPhill  Jan 13, 2018
 #4
avatar+241 
+1

Thanks CPhill!

 
RektTheNoob  Jan 17, 2018
 #2
avatar
0

In triangle ABD, angle BAD =33.2 deg., Angle ABD=47.2 deg. Angle ADB =99.6 deg.
Side AB = 5
Using the Law of Sines, we get:BD=2.777 and AD=3.721, so:
AD^2 =3.721^2 =13.846 ???!!
CPhill: Does this make sense???

 
Guest Jan 13, 2018
 #3
avatar
0

Similarly, in triangle ACD, angle DAC=33.2 deg, angle DCA=66.4 deg, and angle CDA=80.4 deg.

Side AC=4.  By Law of Sines: CD=2.221 and AD=3.718. Therefore:

AD^2 =3.718^2 =13.823 ~ 1,120/81.

 
Guest Jan 14, 2018

20 Online Users

avatar
avatar
avatar
avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details