To find the coordinates of point D that form a parallelogram with points A, B, and C, we need to apply the property of a parallelogram.

In a parallelogram, the diagonals bisect each other. Therefore, the midpoint of diagonal $AC$ should be the same as the midpoint of diagonal $BD$.

Step 1: Calculate the midpoint of diagonal $AC$

Given the coordinates of points A $(-2, -3)$ and C $(-10, 1)$, the midpoint $M_{AC}$ can be calculated as follows:

$M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{-2 + (-10)}{2}, \frac{-3 + 1}{2} \right) = \left( \frac{-12}{2}, \frac{-2}{2} \right) = (-6, -1)$

Step 2: Find the coordinates of point D

Next, we use the property of the midpoint of diagonals. The midpoint of diagonal $BD$ should also be $(-6, -1)$, since it's the same as the midpoint of $AC$. Point B has coordinates $(-4, 8)$, and we can use the midpoint formula again:

$M_{BD} = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) = (-6, -1)$

This gives us two equations:

$\frac{-4 + x_D}{2} = -6 \quad \text{(for the x-coordinate)}$ $\frac{8 + y_D}{2} = -1 \quad \text{(for the y-coordinate)}$

Step 3: Solve for $x_D$ and $y_D$

1. For the x-coordinate: $\frac{-4 + x_D}{2} = -6$ Multiply both sides by 2: $-4 + x_D = -12$ Now, solve for $x_D$: $x_D = -12 + 4 = -8$

2. For the y-coordinate: $\frac{8 + y_D}{2} = -1$ Multiply both sides by 2: $8 + y_D = -2$ Now, solve for $y_D$: $y_D = -2 - 8 = -10$

Step 4: Conclusion

The coordinates of point D are $(-8, -10)$.

Thus, the points $A(-2, -3)$, $B(-4, 8)$, $C(-10, 1)$, and $D(-8, -10)$ form a parallelogram.

Would you like further details on the concept of parallelograms or how we derived the solution? Or perhaps a visual representation of the points on a graph?

Related Questions:

1. How do you prove that diagonals of a parallelogram bisect each other?
2. What properties define a parallelogram?
3. How do you find the area of a parallelogram when given its vertices?
4. Can the parallelogram have any other configurations based on different diagonal selections?
5. What is the relationship between the slopes of opposite sides in a parallelogram?

Tip: When working with parallelograms, always use the midpoint property of diagonals to find unknown points, especially if you have three points already.