In a parallelogram, the diagonals bisect each other at their midpoints. This means that the midpoint $M$ of the diagonals is equidistant from the vertices of the parallelogram. If you're given the coordinates of points $A$, $B$, and $M$, you can use the midpoint property to find the coordinates of points $C$ and $D$.

Step-by-step process:

1. Use the midpoint formula:
   The midpoint $M$ of the diagonals connects points $A$ and $C$, and points $B$ and $D$. The midpoint formula is: $M = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right)$ where $A(x_A, y_A)$ and $C(x_C, y_C)$ are the coordinates of points $A$ and $C$, respectively.

   From this, we can express the coordinates of $C$ in terms of $A$ and $M$: $x_C = 2x_M - x_A$ $y_C = 2y_M - y_A$ Similarly, for $D$, the midpoint formula for the other diagonal (connecting points $B$ and $D$) gives: $M = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right)$ So, you can solve for $D$: $x_D = 2x_M - x_B$ $y_D = 2y_M - y_B$

Summary of the steps:

• To find the coordinates of $C$, use: $C(x_C, y_C) = (2x_M - x_A, 2y_M - y_A)$
• To find the coordinates of $D$, use: $D(x_D, y_D) = (2x_M - x_B, 2y_M - y_B)$

This gives you the coordinates of $C$ and $D$ based on the given points $A$, $B$, and $M$.

Would you like further details or help with an example? Here are some follow-up questions to deepen the understanding:

1. What properties of parallelograms make the diagonals bisect each other at the midpoint?
2. How would you find the area of a parallelogram if the vertices were given as coordinates?
3. How can you use vectors to represent the sides and diagonals of a parallelogram?
4. How can you prove that the diagonals of a parallelogram bisect each other using vector geometry?
5. How does the parallelogram law relate to the bisecting diagonals?

Tip: Always double-check the midpoint calculations and make sure you understand the relationship between the vertices and midpoints. This can prevent mistakes in geometry problems.