# In triangle ABC,  AD is the bisector of the angle A and AD meets the side BC at D. If AB=5.6cm, BC=6cm and BD=3.2cm. What is the length of the side Ac ?

vaaruni | High School Teacher | (Level 1) Salutatorian

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Given : AD is the bisector of the angle A meeting the side BC at point D , AB=5.6cm. , Bc=6cm. and BD=3.2cm.  Let us draw from point D a line (DE) parallel to the side DA meeting AC at point E.

therefore angle DAE=angle ADE  and  Triangle ADE is an isosceles triangle, where the side AE=DE  ,  Let  AE = DE = x

Next in triangle EDC and  triangle ABC :  angle BAC=angle DEC [corrosponding angles equal,BA parallel to DE] , angle BCA=angle ECD [ common angle] , angle ABD=angle EDC [ corrosponding angles equal, BA parallel to DE] .                                      Therefore triangle ABC ~ triangle ADE [test A.A.A] .                  Now Using Propprtionality property of similar triangle We get :

AB/BC = ED/AB =  EC/AC     ->  2.8/6 = x/5.6 =EC/AC  ------(1)

Taking  x/5.6 = 2.8/6   ->  x = (2.8*5.6)/6   --> x = 196/75 ---(2)

Next Taking the ratio :  DC/BC = EC/AC

2.8/6 = EC/AC     --->   2.8/6 =(AC - AE)/AC  [since AC= AE + EC]

2.8/6 = AC/AC - AE/AC   --->  2.8/6 = 1 - AE/AC

OR,    2.8/6 = 1 -x/AC

Or,  x/AC = 1 - 2.8/6 =  (6-2.8)/6 = 3.2/6

Or,  X/AC = 3.2/6

OR,  AC = (6*x)/3.2 = (6*196)/(3.2*75) [ substituting value of x ]

Or,   Ac = (60*196)/(32*75) = 49/10 = 4.9

Hence  AC = 4.9   <--- Answer