In ΔABC, AD is the bisector of ∠A. If BC = 10 cm, BD = 6 cm and AC = 6 cm, find AB.

Construction: Draw a line through C parallel to AD, meeting BA produced at E.

In ΔBCE,

Since AD ∥ CE, by Basic Proportionality Theorem:

$\dfrac{BD}{DC} = \dfrac{AB}{AE}$ …(i)

Also, since AD ∥ CE:

∠BAD = ∠AEC [Corresponding angles are equal]

∠DAC = ∠ACE [Alternate interior angles are equal]

Since AD is bisector of ∠A, ∠BAD = ∠DAC.

Therefore, ∠AEC = ∠ACE.

In ΔACE, sides opposite to equal angles are equal:

AE = AC …(ii)

Substitute (ii) into (i):

$\dfrac{BD}{DC} = \dfrac{AB}{AC}$

Given,

AC = 6 cm

BC = 10 cm

BD = 6 cm

DC = BC - BD [from figure]

DC = 10 - 6 = 4 cm

Let length of AB be x,

$\Rightarrow \dfrac{BD}{DC} = \dfrac{AB}{AC} \\[1em] \Rightarrow \dfrac{6}{4} = \dfrac{x}{6} \\[1em] \Rightarrow \dfrac{3}{2} = \dfrac{x}{6} \\[1em] \Rightarrow \dfrac{3}{2} \times 6 = x \\[1em] \Rightarrow x = 9.$

Hence, length of AB is 9 cm.