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?

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.