`A(4,3,5),B(0,-2,2) and C(3,2,1)` are three points. The coordinates of the point in which the bisector of `angleBAC` meets the side `bar(BC)` is

Options

A. `((15)/(8),(4)/(8),(11)/(8))`
B. `((12)/(7),(2)/(7),(10)/(7))`
C. `((9)/(5),(2)/(5),(7)/(5))`
D. `((3)/(2),0,(3)/(2))`

Answer

To find the coordinates of the point where the bisector of angle BAC meets the side BC, we will follow these steps:

### Step 1: Identify the coordinates of points A, B, and C
Given:
- A(4, 3, 5)
- B(0, -2, 2)
- C(3, 2, 1)

### Step 2: Calculate the lengths of sides AB and AC
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
For AB:
\[ AB = \sqrt{(0 - 4)^2 + (-2 - 3)^2 + (2 - 5)^2} = \sqrt{(-4)^2 + (-5)^2 + (-3)^2} = \sqrt{16 + 25 + 9} = \sqrt{50} = 5\sqrt{2} \]

For AC:
\[ AC = \sqrt{(3 - 4)^2 + (2 - 3)^2 + (1 - 5)^2} = \sqrt{(-1)^2 + (-1)^2 + (-4)^2} = \sqrt{1 + 1 + 16} = \sqrt{18} = 3\sqrt{2} \]

### Step 3: Determine the ratio of lengths AB and AC
The ratio \( \frac{AB}{AC} \) is:
\[ \frac{AB}{AC} = \frac{5\sqrt{2}}{3\sqrt{2}} = \frac{5}{3} \]
This means that the point D divides BC in the ratio 5:3.

### Step 4: Use the section formula to find the coordinates of point D
The section formula states that if a point D divides the line segment joining points B(x1, y1, z1) and C(x2, y2, z2) in the ratio m:n, then the coordinates of D are given by:
\[ D\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]
Here, m = 5, n = 3, and the coordinates of B and C are:
- B(0, -2, 2)
- C(3, 2, 1)

Calculating the coordinates of D:
1. **X-coordinate:**
\[ D_x = \frac{5 \cdot 3 + 3 \cdot 0}{5 + 3} = \frac{15 + 0}{8} = \frac{15}{8} \]

2. **Y-coordinate:**
\[ D_y = \frac{5 \cdot 2 + 3 \cdot (-2)}{5 + 3} = \frac{10 - 6}{8} = \frac{4}{8} = \frac{1}{2} \]

3. **Z-coordinate:**
\[ D_z = \frac{5 \cdot 1 + 3 \cdot 2}{5 + 3} = \frac{5 + 6}{8} = \frac{11}{8} \]

### Final Coordinates of Point D
Thus, the coordinates of point D where the bisector of angle BAC meets side BC are:
\[ D\left(\frac{15}{8}, \frac{1}{2}, \frac{11}{8}\right) \]