Trapezium with sides in ratio — find the smaller parallel side: The area of a trapezium is 384 sq cm. Its parallel sides are in the ratio 3:5, and the perpendicular distance between them is 12 cm. Find the length of the smaller parallel side.

Introduction / Context:A trapezium’s area depends on the average of the parallel sides multiplied by the distance between them. Ratios allow us to parameterize side lengths.

Given Data / Assumptions:

• Area = 384 sq cm
• Parallel sides in ratio 3:5 → let them be 3x and 5x
• Distance (height) = 12 cm

Concept / Approach:Area A = (1/2)*(sum of parallel sides)*height = (1/2)*(3x + 5x)*12 = 48x. Solve 48x = 384, then take the smaller side 3x.

Step-by-Step Solution:48x = 384 ⇒ x = 8Smaller side = 3x = 24 cm

Verification / Alternative check:Parallel sides are 24 cm and 40 cm; area = (1/2)*(64)*12 = 384 ✓

Why Other Options Are Wrong:16 cm and 32 cm mismatch the solved ratio; 40 cm is the larger parallel side.

Common Pitfalls:Using the difference of the sides instead of the sum; misreading which side is smaller.

Final Answer:24 cm