question_answer

Let A(4, 7, 8), B(2, 3, 4), C(2, 5, 7) be the vertices of a triangle ABC. The length of internal bisector of \[\angle A\] is

A) \[\frac{\sqrt{34}}{2}\]

B) \[\frac{3}{2}\sqrt{34}\]

C) \[\frac{2}{3}\sqrt{34}\]

D) \[\frac{\sqrt{34}}{3}\]

Correct Answer: C

Solution :

[c] \[AB=6,\,\,BC=\sqrt{13,}\,\,CA=3\] \[\therefore AB:AC=2:1\] Internal bisector of an angle divides the opposite side in the ratio of adjacent sides \[\therefore \frac{BD}{CD}=\frac{AB}{AC}=\frac{2}{1}\] \[\therefore \]      Coordinate of D are \[\left( 2,\frac{13}{3},6 \right)\] \[\therefore \]      Length \[AD=\frac{2}{3}\sqrt{34}\]