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.