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:
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:
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The angle bisector theorem states:
\[
\frac{BD}{DC} = \frac{AB}{AC}
\]
Use this ratio to solve for missing segment lengths.
Updated On: Oct 27, 2025
- \( 2.5 \) cm
- \( 3.5 \) cm
- \( 4.5 \) cm
- \( 4 \) cm
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The Correct Option is B
Solution and Explanation
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 \( x \):
\[ 7x = 5(6 - x) \] \[ 7x = 30 - 5x \] \[ 12x = 30 \] \[ x = 2.5, \quad DC = 6 - 2.5 = 3.5 \]
\[ \frac{BD}{DC} = \frac{AB}{AC} = \frac{10}{14} = \frac{5}{7} \] Since \( BC = 6 \) cm, let \( BD = x \) and \( DC = 6 - x \).
\[ \frac{x}{6-x} = \frac{5}{7} \] Solving for \( x \):
\[ 7x = 5(6 - x) \] \[ 7x = 30 - 5x \] \[ 12x = 30 \] \[ x = 2.5, \quad DC = 6 - 2.5 = 3.5 \]
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