# In the adjoining figure, if AD is the bisector of ∠A, what is AC?

Question:

In the adjoining figure, if AD is the bisector of ∠A, what is AC?

Solution:

GIVEN: $A B=6 \mathrm{~cm}, B D=3 \mathrm{~cm}$ and $D C=2 \mathrm{~cm} .$ Also, $A D$ is the bisector of $\angle A$.

TO FIND: AC

SOLUTION: We know that the internal bisector of a triangle divides the opposite side internally in the ratio of the sides containing the angle. Therefore,

$\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{\mathrm{BD}}{\mathrm{DC}}$

$\frac{6}{\mathrm{AC}}=\frac{3}{2}$

$\mathrm{AC}=\frac{6 \times 2}{3}$

$\mathrm{AC}=4 \mathrm{~cm}$