Question:
In \( \triangle ABC \), the internal bisector of \( \angle A \) meets BC at D. If \( AB = 4 \), \( AC = 3 \) and \( \angle A = 60^\circ \), then the length of AD is:
In \( \triangle ABC \), the internal bisector of \( \angle A \) meets BC at D. If \( AB = 4 \), \( AC = 3 \) and \( \angle A = 60^\circ \), then the length of AD is:
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The Angle Bisector Theorem is useful for splitting sides proportionally in triangles.
Updated On: Aug 4, 2025
- \( 2 \sqrt{3} \)
- \( \frac{12 \sqrt{3}}{7} \)
- \( \frac{15 \sqrt{3}}{8} \)
- \( \frac{6 \sqrt{3}}{7} \)
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The Correct Option is B
Solution and Explanation
Using the Angle Bisector Theorem, we know that:
\[
\frac{BD}{DC} = \frac{AB}{AC} = \frac{4}{3}
\]
Let \( BD = 4x \) and \( DC = 3x \). The length of BC is:
\[
BC = BD + DC = 4x + 3x = 7x
\]
Using the formula for the length of the angle bisector \( AD \):
\[
AD^2 = AB \times AC \left(1 - \frac{BC^2}{(AB + AC)^2}\right)
\]
Substituting values:
\[
AD^2 = 4 \times 3 \left(1 - \frac{(7x)^2}{(4 + 3)^2}\right) = 12 \times \left(1 - \frac{49x^2}{49}\right) = 12 \times \left(1 - x^2\right)
\]
Solving this for the exact length of \( AD \), we find \( AD = \frac{12 \sqrt{3}}{7} \). Hence, the Correct Answer is \( \frac{12 \sqrt{3}}{7} \).
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