$\begin{array}{1 1}(A) \large\frac{3\sqrt {34}}{2} \\ (B) \large\frac{2\sqrt {34}}{3} \\(C) \large\frac{\sqrt {34}}{2} \\ (D) \large\frac{\sqrt {34}}{3} \end{array} $

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- The internal angular bisector of angle $A$ divides the side $BC$ in the ratio $AB:AC$

Given: $A(4,7,8),\:B(2,3,4)\:and\:C(2,5,7)$

$\Rightarrow\:AB=\sqrt {4+16+16}=6$

$\Rightarrow\:AC=\sqrt {4+4+1}=3$

Let the angular bisector of $A$ be $AD$, where $D$ is on $BC$.

$\Rightarrow\:D$ divides $BC$ in the ratio $6:3=2:1$

$\therefore $ coordinates of $D$ using section formula is given by

$D\big(\large\frac{4+2}{2+1},\frac{10+3}{2+1},\frac{14+4}{2+1}\big)$$=(2,\large\frac{13}{3}$$,6)$

Now

Th length of the internal angular bisector of angle $A$ is $AD$

$AD=\sqrt {4+64/9+4}=\large\frac{\sqrt {136}}{3}=\frac{2\sqrt {34}}{3}$

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