In triangle ABC, AD is the internal angular bisector of angle at vertex A. If BD = 4cm, DC = 3cm and AB = 6cm. then find AC.
 

(a) 9 cm
(b) 4.5 cm
(c) 3 cm
(d)None

Hint: Use the Angle Bisector theorem, An angle bisector of a triangle will divide the opposite side into two segments that are proportional to the other two sides of the triangle. \[\dfrac{BD}{DC}=\dfrac{AB}{AC}\]

Complete step-by-step answer:
Angle bisector is a line which bisects the internal angle exactly by half.
So from above figure we can say
The value of angle BAD is equal to the value of angle DAC. As the bisector is the ray in the interior of an angle forming two congruent angles.
The AD is an angle bisector.
\[\dfrac{BD}{DC}=\dfrac{AB}{AC}\]
In question it is given:
BD = 4cm, DC = 3cm and AB = 6cm
Substituting above values, we get:
\[\dfrac{4}{3}=\dfrac{6}{AC}\]
By simplifying, we get:
\[AC=\dfrac{18}{4}=\dfrac{9}{2}\]
AC = 4.5cm
Therefore the length of side AC is 4.5 cm.

Note: Apply the angle bisector theorem carefully. Observe that the bisector is from A or B. You should understand both cases. Any kind of calculation mistakes in the cross-multiplication must be avoided.