If the diagonals of a parallelogram ABCD are along the lines `x+5y=7` and `10x-2y=9`, then ABCD must be a

Options

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

Answer

To determine the type of parallelogram ABCD based on the given 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. \( x + 5y = 7 \)
2. \( 10x - 2y = 9 \)

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

For the first line \( x + 5y = 7 \):
\[ 5y = -x + 7 \\ y = -\frac{1}{5}x + \frac{7}{5} \]
Thus, the slope \( m_1 = -\frac{1}{5} \).

For the second line \( 10x - 2y = 9 \):
\[ -2y = -10x + 9 \\ y = 5x - \frac{9}{2} \]
Thus, the slope \( m_2 = 5 \).

### Step 3: Check if the diagonals are perpendicular
To check if the diagonals are perpendicular, we multiply the slopes \( m_1 \) and \( m_2 \):
\[ m_1 \cdot m_2 = \left(-\frac{1}{5}\right) \cdot 5 = -1 \]
Since the product of the slopes is \(-1\), the diagonals are perpendicular.

### Step 4: Determine the type of parallelogram
In a parallelogram, if the diagonals are perpendicular, it indicates that the parallelogram is a rhombus. Therefore, the type of parallelogram ABCD must be a rhombus.

### Final Answer
ABCD must be a rhombus.

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