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Pls solve midpoint theorem sum with working 

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

ABCD is a parallelogram. P and Q are midpoints. Prove that PQ = MN = ½ AC.

This problem involves proving that PQ = MN = ½ AC in a parallelogram ABCD where P and Q are midpoints of sides AB and CD. The solution uses the Midpoint Theorem, making it suitable for students in Grades 8-10.

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