Trapezium with intersecting diagonals split ratio 2:9:\nIn a trapezium, one diagonal divides the other in the ratio 2 : 9 at their intersection. If the longer parallel side is 45 cm, find the exact length (in cm) of the other parallel side.

Introduction / Context:
Diagonals of a trapezium intersect and divide each other proportionally to the lengths of the parallel sides. Knowing the division ratio and one base lets us recover the other base length.

Given Data / Assumptions:

• Trapezium with parallel sides (bases) of lengths B_long = 45 cm and B_short = x cm.
• At the intersection of diagonals, one diagonal divides the other in the ratio 2:9.

Concept / Approach:
Standard result: if diagonals AC and BD meet at O, then AO/OC = BO/OD = (length of one base)/(length of the other base). Interpreting the given ratio as AO:OC = 2:9, the smaller:larger relation matches base_short:base_long = 2:9.

Step-by-Step Solution:

Verification / Alternative check:
Had we inverted the ratio, we would get an implausible larger base; the smaller base must be less than 45 cm, consistent with x = 10 cm.

Why Other Options Are Wrong:
5, 14, 18, 22.5 do not satisfy x/45 = 2/9.

Common Pitfalls:
Confusing which base corresponds to the smaller ratio part; always map the smaller division to the shorter base in a trapezium.

Final Answer:
10