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\[

A.\dfrac{1}{8}\sqrt {170} \\

B.\dfrac{3}{8}\sqrt {170} \\

C.\dfrac{5}{8}\sqrt {170} \\

D.\dfrac{7}{8}\sqrt {170} \\

\]

Answer
Verified

In triangle ABC, the internal bisector of angle A will meet BC at D. As from the geometry, we know that any angle bisector in the triangle will bisect the opposite side in the ratio of the other two sides.

Thus, according to angle bisector theorem, we have

$\dfrac{{AB}}{{AC}} = \dfrac{{BD}}{{CD}}$…(1)

Distance formula for points\[({x_1},{y_1},{z_1}) and ({x_2},{y_2},{z_2})\] is:

\[\sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2} + {{({z_1} - {z_2})}^2}} \]

Since coordinates of A , B and C are (5,4,6), (1,−1,3) and (4,3,2). We will calculate lengths of AB and AC by using distance formula as follows.

$

AB = \sqrt {{{(1 - 5)}^2} + {{( - 1 - 4)}^2} + {{(3 - 6)}^2}} \\

= \sqrt {16 + 25 + 9} \\

= 5\sqrt 2 \\

$

$

AC = \sqrt {{{(4 - 5)}^2} + {{(3 - 4)}^2} + {{(2 - 6)}^2}} \\

= \sqrt {1 + 1 + 16} \\

= 3\sqrt 2 \\

$

Thus from equation (1) , we get

$

\dfrac{{AB}}{{AC}} = \dfrac{{BD}}{{CD}} = \dfrac{{5\sqrt 2 }}{{3\sqrt 2 }} \\

\Rightarrow \dfrac{{BD}}{{CD}} = \dfrac{5}{3} \\

$….(2)

Now, point D will divide the BC in a ratio of 5:3.

So, by section formula,

$

x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}} \\

y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}} \\

z = \dfrac{{m{z_2} + n{z_1}}}{{m + n}} \\

$

Putting coordinates of B and C with m=5 and n=3 in above formula, we get

$

x = \dfrac{{5 \times 4 + 3 \times 1}}{{5 + 3}} = \dfrac{{23}}{8} \\

y = \dfrac{{5 \times 3 - 3 \times 1}}{{5 + 3}} = \dfrac{{12}}{8} \\

z = \dfrac{{5 \times 2 + 3 \times 3}}{{5 + 3}} = \dfrac{{19}}{8} \\

$

So coordinate of D is $(\dfrac{{23}}{8},\dfrac{{12}}{8},\dfrac{{19}}{8})$. Now, we will find the distance between A and D, to get length of AD by using distance formula as follows,

\[

AD = \sqrt {{{(5 - \dfrac{{23}}{8})}^2} + {{(4 - \dfrac{{12}}{8})}^2} + {{(6 - \dfrac{{19}}{8})}^2}} \\

= \sqrt {{{(\dfrac{{17}}{8})}^2} + {{(\dfrac{{20}}{8})}^2} + {{(\dfrac{{29}}{8})}^2}} \\

= \dfrac{1}{8}\sqrt {1530} \\

= \dfrac{3}{8}\sqrt {170} \\

\]

Thus the length of AD will be $\dfrac{3}{8}\sqrt {170} $.