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 will plot the points \( A \), \( B \), and \( C \) on a graph paper.
- Point \( A(-2, 2) \) is located 2 units up on the y-axis and 2 units left on the x-axis.
- Point \( B(8, 2) \) is located 2 units up on the y-axis and 8 units right on the x-axis.
- Point \( C(4, -4) \) is located 4 units down on the y-axis and 4 units right on the x-axis.
### Step 2: Draw the Parallelogram
Next, we will connect the points \( A \), \( B \), and \( C \) to visualize the parallelogram.
- Draw a line segment from \( A \) to \( B \).
- Draw a line segment from \( A \) to \( C \).
- Draw a line segment from \( B \) to \( C \).
### Step 3: Find the Fourth Vertex \( D \)
To find the coordinates of the fourth vertex \( D \), we can use the property of the diagonals of a parallelogram. The diagonals bisect each other.
1. **Calculate the midpoint of diagonal \( AC \)**:
\[
\text{Midpoint of } AC = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{-2 + 4}{2}, \frac{2 + (-4)}{2} \right) = \left( \frac{2}{2}, \frac{-2}{2} \right) = (1, -1)
\]
2. **Let the coordinates of \( D \) be \( (x_D, y_D) \)**. The midpoint of diagonal \( BD \) must also equal \( (1, -1) \):
\[
\text{Midpoint of } BD = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) = \left( \frac{8 + x_D}{2}, \frac{2 + y_D}{2} \right)
\]
3. **Set the midpoints equal**:
\[
\frac{8 + x_D}{2} = 1 \quad \text{and} \quad \frac{2 + y_D}{2} = -1
\]
4. **Solve for \( x_D \)**:
\[
8 + x_D = 2 \implies x_D = 2 - 8 = -6
\]
5. **Solve for \( y_D \)**:
\[
2 + y_D = -2 \implies y_D = -2 - 2 = -4
\]
### Step 4: Conclusion
The coordinates of the fourth vertex \( D \) are \( (-6, -4) \).
