Let `P=(-1,0),Q=(0,0)` and `R=(3,3sqrt(3))` be three points. The equation of the bisector of the angle PQR is

Options

A. `sqrt(3)x+y=0`
B. `x+(sqrt(3))/2y=0`
C. `(sqrt(3))/2x+y=0`
D. `x+sqrt(3)y=0`

Answer

To find the equation of the angle bisector of the angle PQR, we will follow these steps:

### Step 1: Identify the Points
We have three points:
- \( P = (-1, 0) \)
- \( Q = (0, 0) \)
- \( R = (3, 3\sqrt{3}) \)

### Step 2: Calculate the Slopes
We need to find the slopes of the lines \( PQ \) and \( QR \).

1. **Slope of \( PQ \)**:
\[ \text{slope of } PQ = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 0}{0 - (-1)} = 0 \]
The slope of line \( PQ \) is \( 0 \) (horizontal line).

2. **Slope of \( QR \)**:
\[ \text{slope of } QR = \frac{3\sqrt{3} - 0}{3 - 0} = \frac{3\sqrt{3}}{3} = \sqrt{3} \]

### Step 3: Calculate the Angles
The angle \( \angle PQR \) can be calculated from the slopes:
- The angle \( \theta_1 \) for slope \( 0 \) (line \( PQ \)) is \( 0^\circ \).
- The angle \( \theta_2 \) for slope \( \sqrt{3} \) (line \( QR \)) is \( 60^\circ \).

### Step 4: Find the Angle Bisector
The angle bisector divides the angle \( \angle PQR \) into two equal parts. Therefore, the angle of the bisector is:
\[ \text{Angle of bisector} = \frac{0^\circ + 60^\circ}{2} = 30^\circ \]

### Step 5: Calculate the Slope of the Bisector
The slope of the angle bisector can be calculated using:
\[ \text{slope of bisector} = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

### Step 6: Write the Equation of the Bisector
Using point \( Q(0, 0) \) and the slope \( \frac{1}{\sqrt{3}} \), we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (0, 0) \) and \( m = \frac{1}{\sqrt{3}} \):
\[ y - 0 = \frac{1}{\sqrt{3}}(x - 0) \]
This simplifies to:
\[ y = \frac{1}{\sqrt{3}}x \]

### Step 7: Rearranging the Equation
To express the equation in standard form, we can multiply through by \( \sqrt{3} \):
\[ \sqrt{3}y - x = 0 \]
or
\[ x - \sqrt{3}y = 0 \]

### Final Answer
The equation of the bisector of the angle \( PQR \) is:
\[ x - \sqrt{3}y = 0 \]