First slide

Introduction to 3-D Geometry

Question

$A (5, 4, 6), B (1, - 1, 3) and C (4, 3, 2) form △ A B C . If the internal bisector of angle A meets B C in D,$
$then the length of \bar{A D} is$

Moderate

A

$\frac{1}{8} \sqrt{170}$
B

$\frac{3}{8} \sqrt{170}$
C

$\frac{5}{8} \sqrt{170}$
D

$\frac{7}{8} \sqrt{170}$

Solution

$G i v e n A (5, 4, 6), B (1, - 1, 3) and C (4, 3, 2) \Rightarrow A B = c = \sqrt{50}, B C = a = \sqrt{26} a n d C A = b = \sqrt{18} ∴ l e n g t h o f t h e i n t e r n a l a n g l e b i s e c t o r t h r o u g h A = \sqrt{b c (1 - \frac{a^{2}}{{(b + c)}^{2}})} = \frac{3}{8} \sqrt{170}$

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