In the given problem, PQRS is a trapezium where PQ is parallel to RS, and $m$ and $n$ are midpoints of sides PS and QR, respectively. We are asked to find the length of the segment $MN$, where $M$ and $N$ are the midpoints.

Solution:

For a trapezium, if we take the midpoints of the non-parallel sides (PS and QR in this case), the segment joining these midpoints (MN) is parallel to the parallel sides (PQ and RS) and its length is given by the average of the lengths of PQ and RS.

The formula to calculate the length of MN is: $MN = \frac{PQ + RS}{2}$

Given:

• $PQ = 7 \, \text{cm}$
• $RS = 3 \, \text{cm}$

Now, substitute the values into the formula: $MN = \frac{7 + 3}{2} = \frac{10}{2} = 5 \, \text{cm}$

Thus, the length of $MN$ is 5 cm.

Would you like a further breakdown or explanation of this?

Here are 5 related questions you can explore:

1. What is the definition of a trapezium and its properties?
2. How do you derive the formula for the segment joining midpoints in a trapezium?
3. What happens if $PQ$ and $RS$ are equal?
4. How would the formula change if the trapezium were isosceles?
5. Can you find the area of trapezium PQRS using the given dimensions?

Tip: The midpoint theorem is a powerful tool in geometry, especially in quadrilaterals and triangles, for finding relations between segments.