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We are given that ABCD is a parallelogram, and we need to find the coordinates of point D. We have the following information:

- \( A(-11, 23) \)
- \( B(-13, 29) \)
- \( C(5, 22) \)

To find the coordinates of point D, we can use the property of parallelograms: the diagonals bisect each other. This means the midpoint of diagonal AC is the same as the midpoint of diagonal BD.

Let's find the midpoint of diagonal AC:

\[
\text{Midpoint of AC} = \left( \frac{-11 + 5}{2}, \frac{23 + 22}{2} \right) = \left( \frac{-6}{2}, \frac{45}{2} \right) = (-3, 22.5)
\]

Now, let the coordinates of point D be \( (x, y) \). Using the fact that the midpoint of BD must also be \( (-3, 22.5) \), we can set up the midpoint formula for BD:

\[
\left( \frac{-13 + x}{2}, \frac{29 + y}{2} \right) = (-3, 22.5)
\]

Now, solve for \( x \) and \( y \):

1. \( \frac{-13 + x}{2} = -3 \)
   
   Multiplying both sides by 2:

   \[
   -13 + x = -6 \quad \Rightarrow \quad x = 7
   \]

2. \( \frac{29 + y}{2} = 22.5 \)
   
   Multiplying both sides by 2:

   \[
   29 + y = 45 \quad \Rightarrow \quad y = 16
   \]

Thus, the coordinates of point D are \( (7, 16) \).

Would you like further details, or do you have any other questions?

Here are 5 related questions to deepen your understanding:
1. How can you confirm that opposite sides of a parallelogram are parallel using vectors?
2. How can the area of parallelogram ABCD be calculated using the vertices' coordinates?
3. Can you apply this method to find a missing vertex of a trapezoid or other quadrilaterals?
4. How would you find the equation of the line passing through two points, like A and B?
5. What are some other properties of parallelograms that can help solve geometric problems?

**Tip**: In geometry, the midpoint formula is an essential tool for solving problems involving bisected diagonals, line segments, or centroids.

ABCD is a parallelogram. What are the coordinates of point D? Given points: A(-11, 23), B(-13, 29), and C(5, 22).

Learn how to find the missing vertex of a parallelogram when given three points. This solution uses the midpoint formula and the property of parallelograms that diagonals bisect each other. Example points A(-11, 23), B(-13, 29), and C(5, 22) are used to calculate the coordinates of point D.

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