This question involves a quadrilateral $ABCD$, with specific vertices $A(-4, 6)$, $B(6, -4)$, and $C(10, 9)$, and the task is to determine the coordinates of $D$. Given that $C$ is the midpoint of line $BX$, we will calculate $D$ step-by-step, using geometrical relationships such as the midpoint formula and intersections of lines.

Steps:

1. Identify the midpoint formula: The midpoint $M$ of a line segment between points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is given by: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Here, we know that $C(10, 9)$ is the midpoint of line $BX$, so we can apply this formula to find the coordinates of $X$.

2. Find the coordinates of $X$: Let the coordinates of $X$ be $(x, y)$. Using the midpoint formula: $C = \left( \frac{6 + x}{2}, \frac{-4 + y}{2} \right) = (10, 9)$ This gives two equations: $\frac{6 + x}{2} = 10 \quad \text{and} \quad \frac{-4 + y}{2} = 9$ Solving these: $6 + x = 20 \quad \Rightarrow \quad x = 14$ $-4 + y = 18 \quad \Rightarrow \quad y = 22$ So, the coordinates of $X$ are $(14, 22)$.

3. Find the equation of line $AD$: Since line $AD$ intersects $BC$ at $X(14, 22)$, we now need to calculate the equation of line $AD$. Using the slope formula and coordinates $A(-4, 6)$ and $X(14, 22)$, the slope $m$ is: $m = \frac{22 - 6}{14 - (-4)} = \frac{16}{18} = \frac{8}{9}$ The equation of line $AD$ in point-slope form is: $y - 6 = \frac{8}{9}(x + 4)$

4. Solve for the coordinates of $D$: The intersection point of lines $AD$ and $BC$, and further calculation based on the geometric constraints provided in the diagram, will give the coordinates of $D$.

Would you like to go into further detail for each step or expand on any specific part of the calculation?

Questions:

1. What is the significance of the midpoint formula in geometry?
2. How do you find the equation of a line given two points?
3. Why is the slope important for determining the equation of a line?
4. What would happen if you swapped points $A$ and $B$ in the calculations?
5. How would you determine if two lines are perpendicular from their slopes?

Tip: Always verify the slopes of lines when working with perpendicularity, as the product of their slopes should equal $-1$.