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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 \(-1\).

Find the coordinates of D given quadrilateral ABCD with vertices A(-4,6), B(6,-4), C(10,9), and D, where C is the midpoint of BX. Calculate using geometric principles.

Learn how to calculate the coordinates of point D in quadrilateral ABCD, using key geometry concepts like the midpoint formula and line equations. This example uses vertices A(-4,6), B(6,-4), and C(10,9), solving for D step-by-step. Ideal for students in Grades 10-12.

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