If the coordinates of the vertices of a $ riangle A B C$ are $A(7,6,4), B(5,4,6), C(3,2,0)$ and the bisector of $\angle B A C$ meets the side $B C$ at $D$, then the coordinates of $D$ are

Question

If the coordinates of the vertices of a $	riangle A B C$ are $A(7,6,4), B(5,4,6), C(3,2,0)$ and the bisector of $\angle B A C$ meets the side $B C$ at $D$, then the coordinates of $D$ are

Aicharya · Accepted Answer

$\because A D$ is the angle bisector of $\angle A$.

$\Rightarrow \quad \frac{A B}{A C}=\frac{B D}{C D}$ Now, $\begin{aligned} & A B=\sqrt{4+4+4}=2 \sqrt{3} \\ & A C=\sqrt{16+16+16}=4 \sqrt{3}\end{aligned}$ $\Rightarrow \quad \frac{B D}{C D}=\frac{2 \sqrt{3}}{4 \sqrt{3}}=\frac{1}{2}$ $\Rightarrow D$ divides $B C$ in the ratio $1: 2$ internally. $\Rightarrow \quad D \equiv\left(\frac{3+10}{3}, \frac{8+2}{3}, \frac{12+0}{3}\right)$ $\equiv\left(\frac{13}{3}, \frac{10}{3}, 4\right)$

If the coordinates of the vertices of a $\triangle A B C$ are $A(7,6,4), B(5,4,6), C(3,2,0)$ and the…

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