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The problem asks to find the coordinates of point \( D \) in parallelogram \( ABCD \) with the given vertices \( A(-4,0) \), \( B(-3,3) \), and \( C(2,2) \).

### Approach:

In a parallelogram, opposite sides are parallel and equal in length, so the diagonals bisect each other. This property can help find point \( D \).

1. Let's denote the midpoint of diagonal \( AC \) and diagonal \( BD \).
2. The midpoint of both diagonals must be the same.

#### Step 1: Calculate the midpoint of \( AC \)
- The midpoint formula for two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
  \[
  M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
  \]
- Applying this to points \( A(-4, 0) \) and \( C(2, 2) \):
  \[
  M_{AC} = \left( \frac{-4 + 2}{2}, \frac{0 + 2}{2} \right) = (-1, 1)
  \]

#### Step 2: Set the midpoint of \( BD \) equal to \( M_{AC} \)
- Let the coordinates of point \( D \) be \( (x_D, y_D) \).
- The midpoint of \( BD \) is:
  \[
  M_{BD} = \left( \frac{-3 + x_D}{2}, \frac{3 + y_D}{2} \right)
  \]
- Since \( M_{AC} = M_{BD} \), equate the coordinates:
  \[
  \frac{-3 + x_D}{2} = -1 \quad \text{and} \quad \frac{3 + y_D}{2} = 1
  \]

#### Step 3: Solve the system of equations
- From \( \frac{-3 + x_D}{2} = -1 \):
  \[
  -3 + x_D = -2 \quad \Rightarrow \quad x_D = 1
  \]
- From \( \frac{3 + y_D}{2} = 1 \):
  \[
  3 + y_D = 2 \quad \Rightarrow \quad y_D = -1
  \]

#### Final Answer:
The coordinates of point \( D \) are \( (1, -1) \).

Would you like any additional details or explanations on this?

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Here are 5 related questions:
1. How can we prove the diagonals of a parallelogram bisect each other?
2. Can we use vector methods to solve this problem?
3. How do you determine if four points form a parallelogram?
4. What are the properties of diagonals in special parallelograms like rectangles or rhombuses?
5. How do we find the area of a parallelogram given its vertices?

**Tip:** When solving geometry problems, sketching the figure often provides helpful visual insights.

If ABCD is a parallelogram with vertices at A(-4,0), B(-3,3), and C(2,2), find the coordinates of point D.

Learn how to find the missing vertex D in a parallelogram ABCD with given vertices A(-4,0), B(-3,3), and C(2,2). This problem involves the properties of parallelograms and the midpoint formula, suitable for students in Grades 8-10.

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