The diagonals of a parallelogram PQRS are along the lines x+3y=4 and 6x-2y=7. Then PQRS must be a

Options

A. Rectangle
B. Square
C. Cyclic Quadrilateral
D. Rhombus

Answer

To determine the type of parallelogram PQRS based on the given diagonals, we can follow these steps:

### Step 1: Identify the equations of the diagonals
The diagonals of the parallelogram are given by the equations:
1. \( L_1: x + 3y = 4 \)
2. \( L_2: 6x - 2y = 7 \)

### Step 2: Find the slopes of the lines
To find the slopes of the lines, we need to rewrite the equations in the slope-intercept form \( y = mx + b \).

For the first line \( L_1: x + 3y = 4 \):
- Rearranging gives us:
\[ 3y = -x + 4 \implies y = -\frac{1}{3}x + \frac{4}{3} \]
- The slope \( m_1 \) of line \( L_1 \) is \( -\frac{1}{3} \).

For the second line \( L_2: 6x - 2y = 7 \):
- Rearranging gives us:
\[ -2y = -6x + 7 \implies y = 3x - \frac{7}{2} \]
- The slope \( m_2 \) of line \( L_2 \) is \( 3 \).

### Step 3: Calculate the product of the slopes
Next, we calculate the product of the slopes \( m_1 \) and \( m_2 \):
\[ m_1 \times m_2 = -\frac{1}{3} \times 3 = -1 \]

### Step 4: Interpret the result
Since the product of the slopes \( m_1 \) and \( m_2 \) is \( -1 \), this indicates that the diagonals of the parallelogram are perpendicular to each other.

### Step 5: Determine the type of parallelogram
In a parallelogram, if the diagonals are perpendicular, it can either be a rhombus or a square. However, we do not have information about the angles of the parallelogram. Therefore, we cannot conclude that it is a square.

Since the diagonals bisect each other at right angles, we can conclude that the parallelogram PQRS is a **rhombus**.

### Final Answer
The parallelogram PQRS must be a **rhombus**.

---