Question

In \( \triangle ABC \), \( AD \) is the bisector of \( \angle ABC \), \( AB = 4 \, \text{cm} \), \( AC = 6 \, \text{cm} \), and \( BD = 2 \, \text{cm} \). Find the value of \( DC \).

• A. 3 cm
• B. 6 cm
• C. 7 cm
• D. 4 cm

Answer 1

The Correct Option is A

By the Angle Bisector Theorem, we know that the ratio of the two segments formed by the angle bisector on the opposite side is equal to the ratio of the other two sides: \[ \frac{AB}{AC} = \frac{BD}{DC} \] Substituting the known values: \[ \frac{4}{6} = \frac{2}{DC} \] Now, cross multiply to find \( DC \): \[ 4 \times DC = 6 \times 2 \] \[ 4 \times DC = 12 \] \[ DC = \frac{12}{4} = 3 \] Thus, the length of \( DC \) is \( \mathbf{3 \, \text{cm}} \). Correct Answer: (A) \( 3 \, \text{cm} \)