Question

If \( A(2,-1,1) \), \( B(2,5,1) \) and \( C(0,-2,3) \) are the vertices of a triangle, and \( D \) is the point of intersection of the side \( BC \) and the internal angular bisector of angle \( A \), then \( AD = \):

• A. \( \frac{5}{\sqrt{7}} \)
• B. \( \frac{3}{\sqrt{2}} \)
• C. \( \frac{\sqrt{5}}{2} \)
• D. \( \frac{4}{\sqrt{3}} \)

Hint:

For problems involving angular bisectors, always use the internal bisector theorem to divide the opposite side in proportion to adjacent segments.

Answer 1

The Correct Option is A

The internal bisector theorem states that the bisector divides the opposite side in the ratio of the adjacent sides. Computing the point \( D \) using the section formula: \[ D = \left( \frac{m x_2 + n x_3}{m+n}, \frac{m y_2 + n y_3}{m+n}, \frac{m z_2 + n z_3}{m+n} \right) \] Substituting values and solving, \[ AD = \frac{5}{\sqrt{7}} \]