In a trapezium, the lengths of the two parallel sides are 8 cm and 4 cm. M and N are the midpoints of the two diagonals of the trapezium. What is the length of segment MN in centimetres?

Introduction / Context:
This is a property based question about a trapezium, a quadrilateral with exactly one pair of parallel sides. It involves the midpoints of the diagonals and a known formula for the segment joining these midpoints. Instead of computing coordinates or angles, you can use a neat geometric result that links the parallel sides directly to the length of the segment between the midpoints of the diagonals.

Given Data / Assumptions:

• The quadrilateral is a trapezium.
• Parallel sides have lengths 8 cm and 4 cm.
• M and N are the midpoints of the two diagonals.
• We need the length of segment MN.
• Standard geometric property: in any trapezium, the segment joining the midpoints of the diagonals has length equal to half the difference of the parallel sides.

Concept / Approach:
For a trapezium with parallel sides a and b (with a greater than b), there is a known result that the line segment connecting the midpoints of its diagonals has length (1/2) * |a - b|. This comes from vector geometry or coordinate geometry analysis of diagonals and midpoints. Using this direct formula allows you to avoid complicated calculations and quickly compute MN from the given side lengths.

Step-by-Step Solution:
Let the lengths of the parallel sides be a = 8 cm and b = 4 cm. Length of segment MN = (1/2) * |a - b|. Compute difference: |a - b| = |8 - 4| = 4 cm. Hence MN = (1/2) * 4 = 2 cm.

Verification / Alternative check:
Though more involved, this result can be derived using coordinate geometry. Place the trapezium with parallel sides on a horizontal axis, compute the coordinates of diagonal midpoints using average formulas, then use distance formula to find MN. Doing so leads back to the same expression involving half the difference of the parallel sides, which confirms the formula and the numeric answer 2 cm for the present values.

Why Other Options Are Wrong:
6 cm and 12 cm: These are based on adding or taking half the sum of sides instead of half the difference. 1 cm and 4 cm: These correspond to misapplied differences or forgetting the half factor.

Common Pitfalls:
Confusing this formula with the one for the mid segment parallel to the bases of a trapezium, which is half the sum of the parallel sides. Using the sum of the sides instead of the difference. Forgetting to take absolute value, although here the longer side is obvious.

Final Answer:
The length of MN is 2 cm