`A(3,2,0),B(5,3,2),C(-9,6,-3)` are the vertices of a triangle ABC. If the bisector of `/_BAC` meets BC at D, then coordinates of D are

Options

A. `(19/8,57/16,17/16)`
B. `(-19/8,57/16,17/16)`
C. `(19/8,-57/16,17/16)`
D. `(19/8,57/16,-17/16)`

Answer

To find the coordinates of point D, where the angle bisector of angle A meets line segment BC, we can follow these steps:

### Step 1: Identify the coordinates of points A, B, and C
The coordinates are given as:
- A(3, 2, 0)
- B(5, 3, 2)
- C(-9, 6, -3)

### Step 2: Calculate the lengths of sides AB and AC using the distance formula
The distance formula in three dimensions is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

#### Calculate AB:
Using points A(3, 2, 0) and B(5, 3, 2):
\[ AB = \sqrt{(5 - 3)^2 + (3 - 2)^2 + (2 - 0)^2} = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

#### Calculate AC:
Using points A(3, 2, 0) and C(-9, 6, -3):
\[ AC = \sqrt{(-9 - 3)^2 + (6 - 2)^2 + (-3 - 0)^2} = \sqrt{(-12)^2 + 4^2 + (-3)^2} = \sqrt{144 + 16 + 9} = \sqrt{169} = 13 \]

### Step 3: Apply the Angle Bisector Theorem
According to the Angle Bisector Theorem, the ratio of the segments created by the angle bisector on the opposite side is equal to the ratio of the other two sides:
\[ \frac{BD}{DC} = \frac{AB}{AC} = \frac{3}{13} \]

### Step 4: Use the section formula to find the coordinates of D
Let D divide BC in the ratio \(3:13\). The coordinates of D can be calculated using the section formula:
\[ D\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n} \right) \]
where \(m = 3\) (BD) and \(n = 13\) (DC), and the coordinates of B and C are:
- B(5, 3, 2)
- C(-9, 6, -3)

#### Calculate the coordinates of D:
\[ D_x = \frac{3(-9) + 13(5)}{3 + 13} = \frac{-27 + 65}{16} = \frac{38}{16} = \frac{19}{8} \]

\[ D_y = \frac{3(6) + 13(3)}{3 + 13} = \frac{18 + 39}{16} = \frac{57}{16} \]

\[ D_z = \frac{3(-3) + 13(2)}{3 + 13} = \frac{-9 + 26}{16} = \frac{17}{16} \]

### Final Coordinates of D
Thus, the coordinates of point D are:
\[ D\left(\frac{19}{8}, \frac{57}{16}, \frac{17}{16}\right) \]

---