Question

If in \( \triangle ABC \), \( AD \) is the bisector of \( \angle BAC \) and \( AB = 10 \) cm, \( AC = 14 \) cm, \( BC = 6 \) cm, then the value of \( DC \) is:

• A. \( 2.5 \) cm
• B. \( 3.5 \) cm
• C. \( 4.5 \) cm
• D. \( 4 \) cm

Hint:

The angle bisector theorem states: \[ \frac{BD}{DC} = \frac{AB}{AC} \] Use this ratio to solve for missing segment lengths.

Answer 1

The Correct Option is B

Using the angle bisector theorem:
\[ \frac{BD}{DC} = \frac{AB}{AC} = \frac{10}{14} = \frac{5}{7} \] Since \( BC = 6 \) cm, let \( BD = x \) and \( DC = 6 - x \).
\[ \frac{x}{6-x} = \frac{5}{7} \] Solving for \( x \):
\[ 7x = 5(6 - x) \] \[ 7x = 30 - 5x \] \[ 12x = 30 \] \[ x = 2.5, \quad DC = 6 - 2.5 = 3.5 \]