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Let's solve the first question (33) from the image.

### Problem 33: Parallelogram with vertices
Given:
- Adjacent vertices of parallelogram \( ABCD \) are \( A(-4, -2) \) and \( B(1, -2) \).
- The diagonals of the parallelogram intersect at point \( M\left(\frac{1}{2}, 0\right) \).

We are required to find the other two vertices of the parallelogram.

#### Key Concept:
- The diagonals of a parallelogram bisect each other, meaning the midpoint of both diagonals is the same. Therefore, the midpoint of diagonal \( AC \) is equal to the midpoint of diagonal \( BD \), which is given as \( M \left(\frac{1}{2}, 0\right) \).

### Step 1: Find vertex \( C \)
The midpoint of diagonal \( AC \) can be calculated as:
\[
M_{AC} = \left( \frac{x_C + x_A}{2}, \frac{y_C + y_A}{2} \right)
\]
Since \( A(-4, -2) \) and \( M \left(\frac{1}{2}, 0\right) \), we equate the midpoint formula to \( M \)'s coordinates:
\[
\left( \frac{x_C - 4}{2}, \frac{y_C - 2}{2} \right) = \left( \frac{1}{2}, 0 \right)
\]
From the first equation:
\[
\frac{x_C - 4}{2} = \frac{1}{2} \quad \Rightarrow \quad x_C - 4 = 1 \quad \Rightarrow \quad x_C = 5
\]
From the second equation:
\[
\frac{y_C - 2}{2} = 0 \quad \Rightarrow \quad y_C - 2 = 0 \quad \Rightarrow \quad y_C = 2
\]
Thus, vertex \( C(5, 2) \).

### Step 2: Find vertex \( D \)
Similarly, for diagonal \( BD \), the midpoint is:
\[
M_{BD} = \left( \frac{x_D + x_B}{2}, \frac{y_D + y_B}{2} \right)
\]
Substitute \( B(1, -2) \) and \( M \left(\frac{1}{2}, 0\right) \):
\[
\left( \frac{x_D + 1}{2}, \frac{y_D - 2}{2} \right) = \left( \frac{1}{2}, 0 \right)
\]
From the first equation:
\[
\frac{x_D + 1}{2} = \frac{1}{2} \quad \Rightarrow \quad x_D + 1 = 1 \quad \Rightarrow \quad x_D = 0
\]
From the second equation:
\[
\frac{y_D - 2}{2} = 0 \quad \Rightarrow \quad y_D - 2 = 0 \quad \Rightarrow \quad y_D = 2
\]
Thus, vertex \( D(0, 2) \).

### Final Answer:
The coordinates of the other two vertices of the parallelogram are:
- \( C(5, 2) \)
- \( D(0, 2) \)

Would you like a detailed explanation for any other question or specific step?

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Here are five related questions for further practice:
1. If the diagonals of a parallelogram bisect each other, what other properties can you deduce about its angles?
2. How do you find the length of the diagonals in a parallelogram if all vertices are given?
3. If a parallelogram has two vertices on the x-axis, what special conditions hold for the vertices on the opposite diagonal?
4. How can you determine whether a quadrilateral is a parallelogram given only its vertex coordinates?
5. What formula do you use to calculate the area of a parallelogram given its vertices?

**Tip:** Remember that the diagonals of a parallelogram always bisect each other, which is a key property in many geometric proofs.

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.

Learn how to determine the coordinates of the remaining vertices of a parallelogram when two adjacent vertices and the intersection of the diagonals are given. This problem uses the midpoint formula and the properties of diagonals in a parallelogram. Ideal for Grade 10 geometry students.

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