In a ΔABC, AB = 10 cm, AC = 14 cm and BC = 6 cm. If AD is the internal bisector of ∠A, then CD is equal to:

1. 3.5 cm

2. 4.8 cm

3. 7 cm

4. 10.5 cm

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

Since AD ∥ EC and BE is the transversal:

∠BAD = ∠AEC [Corresponding angles are equal]

Since AD ∥ EC and AC is the transversal:

∠DAC = ∠ACE [Alternate interior angles]

Given AD is the bisector of ∠A:

∠BAD = ∠DAC

∴ ∠AEC = ∠ACE

In ΔACE, sides opposite to equal angles are equal.

∴ AE = AC = 14 cm

Let DC be x,

Now, in ΔBCE, we have AD ∥ EC. By Basic Proportionality Theorem,

$\Rightarrow \dfrac{BD}{DC} = \dfrac{AB}{AE} \\[1em] \Rightarrow \dfrac{BD}{DC} = \dfrac{10}{14} \\[1em] \Rightarrow \dfrac{BD}{DC} = \dfrac{10}{14} \\[1em] \Rightarrow \dfrac{BD}{DC} = \dfrac{5}{7} \\[1em]$

Let CD = x.

Then BD = BC - CD = 6 - x.

Substituting these values into the ratio:

$\Rightarrow \dfrac{6 - x}{x} = \dfrac{5}{7} \\[1em] \Rightarrow 7(6 - x) = 5x \\[1em] \Rightarrow 42 - 7x = 5x \\[1em] \Rightarrow 42 = 5x + 7x \\[1em] \Rightarrow 12x = 42 \\[1em] \Rightarrow x = \dfrac{42}{12} \\[1em] \Rightarrow x = 3.5 \text{ cm.}$

Thus, CD = 3.5 cm.

Hence, option 1 is the correct option.