Question

A, B, C are the vertices of a triangle ABC. If the bisector of \( \angle BAC \) intersects the side BC at D\((p,q,r)\), then \( \sqrt{2p+q+r} = ? \)

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

The angle bisector theorem is useful for solving geometric problems involving triangles and bisectors.
- Remember the relationship between the angle bisector and the proportional division of the opposite side in the triangle.

Answer 1

The Correct Option is C

We are given the vertices \( A(3,2,-1) \), \( B(4,1,0) \), and \( C(2,1,4) \). The bisector of the angle \( \angle BAC \) intersects the side \( BC \) at point \( D(p,q,r) \). The given condition involves calculating the value of \( \sqrt{2p+q+r} \). We use the formula for the internal bisector of a triangle. In this case, we apply the property of the angle bisector theorem, which relates the coordinates of the points. From calculations involving the distances and the coordinates of \( A \), \( B \), and \( C \), we arrive at the value \( \sqrt{2p+q+r} = 3 \). Thus, the correct answer is \( \boxed{3} \).