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To solve this problem, we need to find the value of \( k \) for which \( D \) is the midpoint of \( AC \).

Given:
- Position vectors: 
  - \( \mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \)
  - \( \mathbf{b} = \begin{pmatrix} 3 \\ 5 \end{pmatrix} \)
  - \( \mathbf{c} = \begin{pmatrix} 7 \\ 4 \end{pmatrix} \)
  - \( \mathbf{d} = \begin{pmatrix} 4 \\ k \end{pmatrix} \)

Since \( D \) is the midpoint of \( AC \), the midpoint formula is:

\[
\mathbf{d} = \frac{\mathbf{a} + \mathbf{c}}{2}
\]

Substituting the values of \( \mathbf{a} \) and \( \mathbf{c} \):

\[
\mathbf{d} = \frac{\begin{pmatrix} 1 \\ 2 \end{pmatrix} + \begin{pmatrix} 7 \\ 4 \end{pmatrix}}{2} = \frac{\begin{pmatrix} 1 + 7 \\ 2 + 4 \end{pmatrix}}{2} = \frac{\begin{pmatrix} 8 \\ 6 \end{pmatrix}}{2} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}
\]

Thus, the position vector of \( D \) is \( \begin{pmatrix} 4 \\ 3 \end{pmatrix} \).

Since \( \mathbf{d} = \begin{pmatrix} 4 \\ k \end{pmatrix} \), we equate the second components:

\[
k = 3
\]

Therefore, the value of \( k \) is \( \boxed{3} \).

Would you like further details or have any questions?

Here are some related questions that might interest you:

1. How do you find the midpoint of a line segment in 3D space?
2. What are the properties of position vectors in coordinate geometry?
3. How can you determine if a point lies on a line segment between two points?
4. What is the significance of midpoints in vector geometry?
5. How do you calculate the distance between two points using their position vectors?

**Tip:** When working with vectors, always ensure that vector operations such as addition and scalar multiplication are performed component-wise for accurate results.

Find the value of k for which D is the midpoint of AC, given the position vectors: a = [1, 2], b = [3, 5], c = [7, 4], and d = [4, k].

Learn how to find the value of k such that point D is the midpoint of line segment AC, using given position vectors. This problem covers key vector concepts and the midpoint formula, suitable for students in Grades 9-10.

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