P (4, 2) and R ( - 2 , 0) are vertices of a rhombus PQRS . What is the equation of diagonal QS ?

Options

A. ` x - 3y = - 2`
B. ` 3 x + y = 4`
C. ` 3 x + y =- 4`
D. ` x - 3 y = 2`

Answer

To find the equation of diagonal QS of rhombus PQRS with given vertices P(4, 2) and R(-2, 0), we can follow these steps:

### Step 1: Identify the coordinates of points P and R
- Given points:
- P = (4, 2)
- R = (-2, 0)

### Step 2: Find the midpoint of diagonal PR
- The midpoint M of a line segment connecting two points (x1, y1) and (x2, y2) is given by the formula:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
- Applying this formula:
\[ M = \left( \frac{4 + (-2)}{2}, \frac{2 + 0}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1) \]

### Step 3: Determine the slope of diagonal PR
- The slope (m) of a line connecting two points (x1, y1) and (x2, y2) is given by:
\[ m = \frac{y2 - y1}{x2 - x1} \]
- For points P(4, 2) and R(-2, 0):
\[ m_{PR} = \frac{0 - 2}{-2 - 4} = \frac{-2}{-6} = \frac{1}{3} \]

### Step 4: Find the slope of diagonal QS
- The diagonals of a rhombus bisect each other at right angles. Therefore, the slope of diagonal QS (m_{QS}) will be the negative reciprocal of the slope of diagonal PR:
\[ m_{QS} = -\frac{1}{m_{PR}} = -3 \]

### Step 5: Use point-slope form to find the equation of diagonal QS
- The point-slope form of a line is given by:
\[ y - y1 = m(x - x1) \]
- Using the midpoint M(1, 1) and slope m_{QS} = -3:
\[ y - 1 = -3(x - 1) \]

### Step 6: Simplify the equation
- Expanding the equation:
\[ y - 1 = -3x + 3 \]
- Rearranging gives:
\[ y = -3x + 4 \]

### Step 7: Write in standard form
- To write the equation in standard form (Ax + By + C = 0):
\[ 3x + y - 4 = 0 \]

Thus, the equation of diagonal QS is:
\[ 3x + y - 4 = 0 \]