First slide

Introduction to 3-D Geometry

Question

$Let A (3, 2, 0), B (5, 3, 2), C (- 9, 6, - 3) are three points forming a triangle. If A D, the$ $bisector of ∠ B A C meets B C in D, then coordinates of D are$

Moderate

A

$(- \frac{19}{8}, \frac{57}{16}, \frac{17}{16})$
B

$(\frac{19}{8}, - \frac{57}{16}, \frac{17}{16})$
C

$(\frac{19}{8}, \frac{57}{16}, - \frac{17}{16})$
D

$(\frac{19}{8}, \frac{57}{16}, \frac{17}{16})$

Solution

$A = (3, 2, 0), B = (5, 3, 2), C (- 9, 6, - 3)$
$A B = \sqrt{4 + 1 + 4} = 3, A C = \sqrt{144 + 16 + 9} = 13$
AD = angular bisector of $∠ B A C$
since BD : DC = 3 :13 $\Rightarrow D = (\frac{- 27 + 65}{16}, \frac{18 + 39}{16}' \frac{- 9 + 26}{16}),$ by section formula
$= (\frac{38}{16}, \frac{57}{16}, \frac{17}{16})$ = $(\frac{19}{8}, \frac{57}{16}, \frac{17}{16})$

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