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

Options

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

Answer

To find the equation of the bisector of the angle PQR formed by the points P=(-1,0), Q=(0,0), and R=(3,3√3), we will follow these steps:

### Step 1: Plot the Points
First, we plot the points P, Q, and R on the Cartesian plane:
- P = (-1, 0)
- Q = (0, 0)
- R = (3, 3√3)

### Step 2: Determine the Angles
Next, we need to find the angles at point Q formed by the lines PQ and QR. The slopes of these lines will help us determine the angles.

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

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

### Step 3: Find the Angle Between the Lines
Using the tangent of the angle between two lines:
\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
where \( m_1 = 0 \) (slope of PQ) and \( m_2 = \sqrt{3} \) (slope of QR).

Calculating:
\[ \tan(\theta) = \left| \frac{0 - \sqrt{3}}{1 + 0 \cdot \sqrt{3}} \right| = \sqrt{3} \]
This means that the angle \( \theta \) is \( 60^\circ \) or \( \frac{\pi}{3} \).

### Step 4: Determine the Angle Bisector
The angle bisector will split the angle at Q into two equal angles of \( 30^\circ \) each. The slope of the angle bisector can be found using the tangent of \( 30^\circ \):
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}}. \]

### Step 5: Equation of the Angle Bisector
The slope of the angle bisector is \( -\frac{1}{\sqrt{3}} \) (since it is in the third quadrant). The equation of the line in point-slope form (passing through Q=(0,0)) is:
\[ y - 0 = -\frac{1}{\sqrt{3}}(x - 0). \]
This simplifies to:
\[ y = -\frac{1}{\sqrt{3}}x. \]

### Step 6: Rearranging the Equation
To express this in standard form:
\[ \sqrt{3}y + x = 0. \]
Thus, the equation of the angle bisector of angle PQR is:
\[ \sqrt{3}x + y = 0. \]

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