In a trapezium, the difference between the lengths of the two parallel sides is 4 cm.\nThe perpendicular distance (height) between these parallel sides is 19 cm.\nIf the area of the trapezium is 475 sq cm, find the lengths (in cm) of the two parallel sides.

Introduction:
This question tests the area formula of a trapezium and solving a pair of linear equations. A trapezium (trapezoid) has one pair of parallel sides, and its area is based on the average of those parallel sides multiplied by the perpendicular distance between them. Here, the area and height are known, and the difference between the parallel sides is known. That naturally produces two equations: one for the sum (from area) and one for the difference (given directly). Solving those two equations gives the two side lengths uniquely.

Given Data / Assumptions:

• Area of trapezium = 475 sq cm
• Height (perpendicular distance) = 19 cm
• Difference between parallel sides = 4 cm
• Let parallel sides be a and b, where a > b
• Trapezium area formula: Area = (1/2) * (a + b) * h

Concept / Approach:
Use the area formula to derive (a + b). Use the given difference to derive (a - b). Then solve by adding and subtracting the equations to isolate a and b. This is a standard “sum and difference” method and avoids trial-and-error.

Step-by-Step Solution:
475 = (1/2) * (a + b) * 19(a + b) = (2 * 475) / 19 = 950 / 19 = 50Given: a - b = 4Add equations: (a + b) + (a - b) = 50 + 42a = 54 => a = 27Then b = 50 - 27 = 23

Verification / Alternative Check:
Check area using a = 27 and b = 23:\nAverage of parallel sides = (27 + 23)/2 = 25.\nArea = 25 * 19 = 475 sq cm, exactly matches the given area. The difference is also 27 - 23 = 4 cm, matching the condition. So the solution is consistent in both constraints.

Why Other Options Are Wrong:
25 and 23 / 24 and 23 / 22 and 23: these do not satisfy (a + b) = 50.28 and 24: difference becomes 4 but sum becomes 52, so area changes.

Common Pitfalls:
Using wrong area formula (like base*height) instead of trapezium average-base formula.Forgetting the 1/2 factor in trapezium area.Mixing up which side is larger when using a - b = 4.Arithmetic mistake in dividing 950 by 19.

Final Answer:
27 cm and 23 cm