If vertices of a parallelogram are respectively `(0,0), (1,0) ,(2,2) and (1,2)` then angle between diagonals is

Options

A. `(pi)/(3)`
B. `(pi)/(2)`
C. `(3pi)/(2)`
D. `(pi)/(4)`

Answer

To find the angle between the diagonals of the parallelogram with vertices at \( A(0,0) \), \( B(1,0) \), \( C(2,2) \), and \( D(1,2) \), we can follow these steps:

### Step 1: Identify the diagonals
The diagonals of the parallelogram are \( AC \) and \( BD \).

### Step 2: Find the coordinates of the diagonals
- Diagonal \( AC \) connects points \( A(0,0) \) and \( C(2,2) \).
- Diagonal \( BD \) connects points \( B(1,0) \) and \( D(1,2) \).

### Step 3: Calculate the slopes of the diagonals
The slope \( m \) of a line connecting two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

#### For diagonal \( AC \):
- \( A(0,0) \) and \( C(2,2) \):
\[ m_{AC} = \frac{2 - 0}{2 - 0} = \frac{2}{2} = 1 \]

#### For diagonal \( BD \):
- \( B(1,0) \) and \( D(1,2) \):
\[ m_{BD} = \frac{2 - 0}{1 - 1} = \frac{2}{0} \]
Since the denominator is zero, the slope \( m_{BD} \) is undefined, indicating that line \( BD \) is vertical.

### Step 4: Use the formula for the angle between two lines
The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Here, \( m_1 = 1 \) (slope of \( AC \)) and \( m_2 \) is undefined (slope of \( BD \)).

### Step 5: Analyze the angle
Since one of the slopes is undefined, it means that one of the lines is vertical. The angle between a vertical line and any line with a defined slope can be calculated as follows:
- The angle between a vertical line and a line with slope \( m \) is given by:
\[ \tan \theta = \left| \frac{1 - m}{1 + m \cdot \infty} \right| = \infty \]
This implies that \( \theta = \frac{\pi}{2} \) radians or \( 90^\circ \).

### Final Answer
The angle between the diagonals \( AC \) and \( BD \) is \( 90^\circ \).

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