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Step-by-Step Solution
Step 1: Assign Position Vectors
Let the position vectors of points A, B, C, and D be
$ \vec{a}, \vec{b}, \vec{c}, $ and $ \vec{d} $ respectively.
Step 2: Find the Midpoints E and F
E is the midpoint of the diagonal AC, so its position vector is:
$$
\vec{E} = \frac{\vec{a} + \vec{c}}{2}.
$$
F is the midpoint of the diagonal BD, so its position vector is:
$$
\vec{F} = \frac{\vec{b} + \vec{d}}{2}.
$$
Step 3: Rewrite the Given Expression in Terms of Position Vectors
We have:
$$
(\overrightarrow{AB} - \overrightarrow{BC})
+ (\overrightarrow{AD} - \overrightarrow{DC})
\;=\; k \,\overrightarrow{FE}.
$$
Using position vectors:
• $ \overrightarrow{AB} = \vec{b} - \vec{a} $
• $ \overrightarrow{BC} = \vec{c} - \vec{b} $
• $ \overrightarrow{AD} = \vec{d} - \vec{a} $
• $ \overrightarrow{DC} = \vec{c} - \vec{d} $
• $ \overrightarrow{FE} = \vec{E} - \vec{F} = \frac{\vec{a} + \vec{c}}{2} - \frac{\vec{b} + \vec{d}}{2}.
$
Step 4: Substitute and Simplify
Substitute these into the left-hand side:
$$
(\vec{b} - \vec{a} - (\vec{c} - \vec{b}))
+ (\vec{d} - \vec{a} - (\vec{c} - \vec{d}))
\;=\; k \Big(\frac{\vec{a} + \vec{c} - \vec{b} - \vec{d}}{2}\Big).
$$
Simplify inside the parentheses:
$$
(\vec{b} - \vec{a} - \vec{c} + \vec{b})
+ (\vec{d} - \vec{a} - \vec{c} + \vec{d})
\;=\; k \Big(\frac{\vec{a} + \vec{c} - \vec{b} - \vec{d}}{2}\Big).
$$
Combine like terms on the left:
$$
(2\vec{b} - \vec{a} - \vec{c}) + (2\vec{d} - \vec{a} - \vec{c})
\;=\; k \,\Big(\frac{\vec{a} + \vec{c} - \vec{b} - \vec{d}}{2}\Big).
$$
Factor out common terms where convenient:
$$
2(\vec{b} + \vec{d} - \vec{a} - \vec{c})
= \frac{k}{2} \big(\vec{a} + \vec{c} - \vec{b} - \vec{d}\big).
$$
Step 5: Solve for k
Observe that:
$$
\vec{b} + \vec{d} - \vec{a} - \vec{c}
= -(\vec{a} + \vec{c} - \vec{b} - \vec{d}).
$$
Hence we can write:
$$
2(\vec{b} + \vec{d} - \vec{a} - \vec{c})
= -\frac{k}{2} (\vec{b} + \vec{d} - \vec{a} - \vec{c}).
$$
Assuming the vector factor is non-zero, canceling it out gives:
$$
2 = -\frac{k}{2}.
$$
Solving for $k$:
$$
k = -4.
$$
Final Answer
The value of $k$ is $-4$.
