Three vertices of a parallelogram ABCD are A = (-2, 2), B = (6, 2) and C = (4.-3). Plot these points on a graph paper and hence use it to find the co ordinates of the fourth vertex D. Also, find the co-ordinates of the mid point of the side CD.

Answer

To find the coordinates of the fourth vertex \( D \) of the parallelogram \( ABCD \) given the vertices \( A(-2, 2) \), \( B(6, 2) \), and \( C(4, -3) \), we can follow these steps:

### Step 1: Plot the Points
1. **Plot the points** \( A(-2, 2) \), \( B(6, 2) \), and \( C(4, -3) \) on a Cartesian coordinate system.
- Point \( A \) is located at \( (-2, 2) \).
- Point \( B \) is located at \( (6, 2) \).
- Point \( C \) is located at \( (4, -3) \).

### Step 2: Identify the Coordinates of Vertex D
2. **Use the properties of a parallelogram** to find the coordinates of vertex \( D \). In a parallelogram, the midpoints of the diagonals are the same. Therefore, we can use the midpoint formula.

The midpoint \( M \) of diagonal \( AC \) can be calculated as:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( A(x_1, y_1) \) and \( C(x_2, y_2) \).

Substituting the coordinates of \( A \) and \( C \):
\[ M_{AC} = \left( \frac{-2 + 4}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{2}{2}, \frac{-1}{2} \right) = (1, -0.5) \]

Now, let \( D(x, y) \) be the coordinates of point \( D \). The midpoint \( M \) of diagonal \( BD \) should also equal \( (1, -0.5) \):
\[ M_{BD} = \left( \frac{6 + x}{2}, \frac{2 + y}{2} \right) \]

Setting \( M_{BD} = (1, -0.5) \):
\[ \frac{6 + x}{2} = 1 \quad \text{and} \quad \frac{2 + y}{2} = -0.5 \]

Solving for \( x \):
\[ 6 + x = 2 \implies x = 2 - 6 = -4 \]

Solving for \( y \):
\[ 2 + y = -1 \implies y = -1 - 2 = -3 \]

Thus, the coordinates of point \( D \) are \( D(-4, -3) \).

### Step 3: Find the Midpoint of Side CD
3. **Calculate the midpoint of side \( CD \)** using the midpoint formula again:
\[ M_{CD} = \left( \frac{x_C + x_D}{2}, \frac{y_C + y_D}{2} \right) \]
Substituting the coordinates of \( C(4, -3) \) and \( D(-4, -3) \):
\[ M_{CD} = \left( \frac{4 + (-4)}{2}, \frac{-3 + (-3)}{2} \right) = \left( \frac{0}{2}, \frac{-6}{2} \right) = (0, -3) \]

### Final Result
- The coordinates of the fourth vertex \( D \) are \( (-4, -3) \).
- The coordinates of the midpoint of side \( CD \) are \( (0, -3) \).