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The problem involves a parallelogram \( ABCD \) with the given adjacent vertices \( A(-4, -2) \) and \( B(1, -2) \). The diagonals intersect at \( M\left(-\frac{1}{2}, 0\right) \), and we need to find the other two vertices \( C \) and \( D \).

### Step-by-step solution:

1. **Midpoint Property of Diagonals**:
   In a parallelogram, the diagonals bisect each other. Thus, the midpoint \( M \) of diagonal \( AC \) is the same as the midpoint of diagonal \( BD \).

   Using this property:
   - The midpoint of \( AC \) is \( M \), and we know \( A \) is \( (-4, -2) \) and \( M \) is \( \left(-\frac{1}{2}, 0\right) \).
   - Similarly, \( M \) is also the midpoint of diagonal \( BD \), where \( B(1, -2) \) is known, and \( D \) is unknown.

2. **Finding Coordinates of \( C \)**:
   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 \( A(-4, -2) \) and \( C(x_C, y_C) \):
   \[
   \left(-\frac{1}{2}, 0\right) = \left( \frac{-4 + x_C}{2}, \frac{-2 + y_C}{2} \right)
   \]
   Solving each component:
   - For the x-coordinate:
     \[
     -\frac{1}{2} = \frac{-4 + x_C}{2} \implies -1 = -4 + x_C \implies x_C = 3
     \]
   - For the y-coordinate:
     \[
     0 = \frac{-2 + y_C}{2} \implies 0 = -2 + y_C \implies y_C = 2
     \]
   So, the coordinates of \( C \) are \( (3, 2) \).

3. **Finding Coordinates of \( D \)**:
   Now, using the same midpoint formula for \( B(1, -2) \) and \( D(x_D, y_D) \):
   \[
   \left(-\frac{1}{2}, 0\right) = \left( \frac{1 + x_D}{2}, \frac{-2 + y_D}{2} \right)
   \]
   Solving each component:
   - For the x-coordinate:
     \[
     -\frac{1}{2} = \frac{1 + x_D}{2} \implies -1 = 1 + x_D \implies x_D = -2
     \]
   - For the y-coordinate:
     \[
     0 = \frac{-2 + y_D}{2} \implies 0 = -2 + y_D \implies y_D = 2
     \]
   So, the coordinates of \( D \) are \( (-2, 2) \).

### Conclusion:
The other two vertices of the parallelogram are:
- \( C(3, 2) \)
- \( D(-2, 2) \)

Would you like a more detailed explanation of any step, or do you have any further questions?

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Here are 5 related questions to explore:

1. How can we verify that diagonals of a parallelogram always bisect each other?
2. Can this method be used to find vertices of other quadrilaterals, like a trapezium?
3. What are the other properties of parallelograms that distinguish them from other quadrilaterals?
4. How would this problem change if the coordinates of one vertex were not given?
5. How can you use vector methods to solve this problem instead of the midpoint formula?

**Tip**: When solving geometry problems with coordinates, visualize the figure and use properties like midpoints, slopes, and distances to find missing values systematically.

The two adjacent vertices of a parallelogram ABCD are A(-4, -2) and B(1, -2). If the diagonals of the parallelogram intersect at the point M(-1/2, 0), then find the other two vertices of the parallelogram.

Discover how to find the missing vertices C and D of a parallelogram given two adjacent vertices and the intersection point of diagonals. Learn about key geometric properties and apply the midpoint formula in this Grade 9-10 level problem.

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