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

Options

A. rectangle
B. kite
C. cyclic quadrilateral
D. rhombus

Answer

To determine the type of parallelogram ABCD based on the given equations of its diagonals, we will follow these steps:

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

### Step 2: Find the slopes of the diagonals
To find the slopes, we will rearrange each equation into the slope-intercept form \( y = mx + b \), where \( m \) is the slope.

**For diagonal AC:**
Starting with the equation \( x + 3y = 4 \):
\[ 3y = -x + 4 \\ y = -\frac{1}{3}x + \frac{4}{3} \]
Thus, the slope \( m_1 \) of diagonal AC is \( -\frac{1}{3} \).

**For diagonal BD:**
Starting with the equation \( 6x - 2y = 7 \):
\[ -2y = -6x + 7 \\ y = 3x - \frac{7}{2} \]
Thus, the slope \( m_2 \) of diagonal BD is \( 3 \).

### Step 3: Check if the diagonals are perpendicular
To check if the diagonals are perpendicular, we multiply their slopes \( m_1 \) and \( m_2 \):
\[ m_1 \cdot m_2 = \left(-\frac{1}{3}\right) \cdot 3 = -1 \]
Since the product of the slopes is \( -1 \), this indicates that the diagonals are perpendicular.

### Step 4: Determine the type of parallelogram
In a parallelogram, if the diagonals are perpendicular, it can be classified as a rhombus. Therefore, parallelogram ABCD must be a rhombus.

### Final Answer:
ABCD must be a rhombus.

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