`A(-2,2),B(8,2)` and `C(4,-4)` are the vertice of a parallelogram ABCD. By plotting the given points on a graph paper, find the co-ordinates of the fourth vertex D.

Explanation

To find the coordinates of the fourth vertex \( D \) of the parallelogram \( ABCD \) given the vertices \( A(-2, 2) \), \( B(8, 2) \), and \( C(4, -4) \), we can follow these steps:

### Step 1: Plot the Points
First, we need to plot the points \( A \), \( B \), and \( C \) on a graph paper.

- **Point A**: \( (-2, 2) \) - This is located 2 units up on the y-axis and 2 units left on the x-axis.
- **Point B**: \( (8, 2) \) - This is located 2 units up on the y-axis and 8 units right on the x-axis.
- **Point C**: \( (4, -4) \) - This is located 4 units down on the y-axis and 4 units right on the x-axis.

### Step 2: Identify the Coordinates of D
In a parallelogram, the diagonals bisect each other. Therefore, we can find the coordinates of point \( D \) using the midpoint formula.

Let the coordinates of point \( D \) be \( (x_D, y_D) \).

The midpoint \( M \) of diagonal \( AC \) can be calculated as follows:

\[ M = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) \]

Substituting the coordinates of \( A \) and \( C \):

\[ M = \left( \frac{-2 + 4}{2}, \frac{2 + (-4)}{2} \right) = \left( \frac{2}{2}, \frac{-2}{2} \right) = (1, -1) \]

### Step 3: Use the Midpoint to Find D
Since \( M \) is also the midpoint of diagonal \( BD \), we can set up the equation:

\[ M = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) \]

Substituting the coordinates of \( B \) and the midpoint \( M \):

\[ (1, -1) = \left( \frac{8 + x_D}{2}, \frac{2 + y_D}{2} \right) \]

From this, we can set up two equations:

1. \( \frac{8 + x_D}{2} = 1 \)
2. \( \frac{2 + y_D}{2} = -1 \)

### Step 4: Solve for \( x_D \) and \( y_D \)

**For \( x_D \)**:
\[ \frac{8 + x_D}{2} = 1 \]
Multiply both sides by 2:
\[ 8 + x_D = 2 \]
Subtract 8 from both sides:
\[ x_D = 2 - 8 = -6 \]

**For \( y_D \)**:
\[ \frac{2 + y_D}{2} = -1 \]
Multiply both sides by 2:
\[ 2 + y_D = -2 \]
Subtract 2 from both sides:
\[ y_D = -2 - 2 = -4 \]

### Step 5: Conclusion
Thus, the coordinates of point \( D \) are \( (-6, -4) \).

### Final Answer
The coordinates of the fourth vertex \( D \) are \( D(-6, -4) \).

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