Question
$$A\left( {3,\,2,\,0} \right),\,B\left( {5,\,3,\,2} \right)$$ and $$C\left( { - 9,\,6,\, - 3} \right)$$ are the vertices of a triangle $$ABC.$$ If the bisector of $$\angle ABC$$ meets $$BC$$ at $$D$$, then coordinates of $$D$$ are :
A.
$$\left( {\frac{{19}}{8},\,\frac{{57}}{{16}},\,\frac{{17}}{{16}}} \right)$$
B.
$$\left( { - \frac{{19}}{8},\,\frac{{57}}{{16}},\,\frac{{17}}{{16}}} \right)$$
C.
$$\left( {\frac{{19}}{8},\, - \frac{{57}}{{16}},\,\frac{{17}}{{16}}} \right)$$
D.
none of these
Answer :
$$\left( {\frac{{19}}{8},\,\frac{{57}}{{16}},\,\frac{{17}}{{16}}} \right)$$
Solution :
$$D$$ divides $$BC$$ in the ratio $$AB : AC$$ i.e. $$3 : 13.$$
Therefore, coordinates of $$D$$ are,
$$\eqalign{
& \left( {\frac{{3 \times - 9 + 13 \times 5}}{{3 + 13}},\,\frac{{3 \times 6 + 13 \times 5}}{{3 + 13}},\,\frac{{3 \times - 3 + 13 \times 2}}{{3 + 13}}} \right) \cr
& {\text{or }}\left( {\frac{{19}}{8},\,\frac{{57}}{{16}},\,\frac{{17}}{{16}}} \right) \cr} $$