The question in the image is asking to find the remaining two vertices of a parallelogram. Here's the breakdown of the problem:

Given:

• Two adjacent vertices of a parallelogram are $A(2, 3)$ and $B(4, 7)$.
• The diagonals of the parallelogram intersect at the point $O(2, 5)$.

We need to find the other two vertices of the parallelogram.

Approach:

1. Midpoint Property: In a parallelogram, the diagonals bisect each other, meaning the midpoint of the diagonal formed by two opposite vertices is the same for both diagonals.

   Since $O(2, 5)$ is the midpoint of diagonals, this property helps us to calculate the unknown vertices.

2. Find the other two vertices (C and D): Let's assume the vertices opposite to $A$ and $B$ are $C(x_1, y_1)$ and $D(x_2, y_2)$ respectively.

   For diagonals $AC$ and $BD$, we know:

   • $O$ is the midpoint of $AC$, so: $O_x = \frac{A_x + C_x}{2}, \quad O_y = \frac{A_y + C_y}{2}$ Using the coordinates of $O(2, 5)$ and $A(2, 3)$, solve for $C(x_1, y_1)$.

   • Similarly, $O$ is the midpoint of $BD$, so: $O_x = \frac{B_x + D_x}{2}, \quad O_y = \frac{B_y + D_y}{2}$ Using the coordinates of $O(2, 5)$ and $B(4, 7)$, solve for $D(x_2, y_2)$.

Let me calculate the exact values of $C$ and $D$.The coordinates of the other two vertices of the parallelogram are:

• $C(2, 7)$
• $D(0, 3)$

This completes the parallelogram with the given vertices $A(2, 3)$, $B(4, 7)$, $C(2, 7)$, and $D(0, 3)$.

Would you like further details or have any other questions?

Here are five related questions to expand your understanding:

1. How can you verify that a given quadrilateral is a parallelogram using coordinate geometry?
2. What properties of parallelograms are useful in solving problems involving diagonals?
3. How would this problem change if you were given three vertices instead of two?
4. What would be the effect on the solution if $O$ were not the midpoint of the diagonals?
5. How can you determine the area of a parallelogram using its vertices' coordinates?

Tip: Always remember that the diagonals of a parallelogram bisect each other, which is a key property used in coordinate geometry.