The problem you've uploaded shows a parallelogram $ABCD$ with diagonals $AC$ and $BD$ intersecting at point $O$. Points $P$ and $Q$ are midpoints of $AB$ and $CD$, respectively. The problem asks to prove that:

$\text{PQ} = \text{MN} = \frac{1}{2} \times \text{AC}$

Here's how to approach the problem:

Step 1: Understand the Midpoint Theorem

The midpoint theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long.

Step 2: Apply the Midpoint Theorem

In parallelogram $ABCD$:

• $P$ and $Q$ are midpoints of $AB$ and $CD$, respectively.
• $M$ and $N$ are midpoints of $AD$ and $BC$, respectively.

Step 3: Proof that PQ = MN = ½ AC

• Consider $\triangle ABD$:
  • $P$ is the midpoint of $AB$ and $M$ is