In the given figure, ABCD is a trapezium where AB // DC. P & Q are the midpoints of AC & BD. Prove that:

Vector PQ= 1/2(Vector AB - Vector DC).

To prove that Vector PQ = 1/2(Vector AB - Vector DC), we can use the midpoint theorem and vector properties.

Here's how you can prove it step by step:

Step 1: Start by drawing the given figure with trapezium ABCD, where AB is parallel to DC, and P and Q are the midpoints of AC and BD, respectively.

Step 2: Label the points: A, B, C, D, P, and Q.

Step 3: Write down the vector definitions for AB, DC, PQ, and AC:
Vector AB = B - A
Vector DC = C - D
Vector PQ = Q - P
Vector AC = C - A

Step 4: Apply the midpoint theorem, which states that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore, we know that AP is parallel to BD and is half its length, and CQ is parallel to BD and is half its length.

Step 5: Express vector PQ using vector AP and vector CQ:
Vector PQ = Vector Q - Vector P

Step 6: Express vector AP in terms of AB and AC:
Vector AP = Vector AC + Vector CB

Step 7: Express vector CQ in terms of DC and CB:
Vector CQ = Vector CB + Vector BD

Step 8: Substitute the values of Vector AP and Vector CQ into Vector PQ:
Vector PQ = (Vector CB + Vector BD) - (Vector AC + Vector CB)

Step 9: Simplify the expression:
Vector PQ = Vector BD - Vector AC

Step 10: Divide both sides of the equation by 2:
Vector PQ = 1/2(Vector BD - Vector AC)

Step 11: Since we know that AB is parallel to DC, and AC is parallel to BD (as AP and CQ are midpoints), we can substitute DC for BD and AB for AC:
Vector PQ = 1/2(Vector AB - Vector DC)

Therefore, we have proven that Vector PQ = 1/2(Vector AB - Vector DC) using the midpoint theorem and vector properties.