In trapezium ABCD, AB is parallel to CD, AB < CD, CD = 6 cm, and the distance between the parallel sides is 4 cm. If the area of trapezium ABCD is 16 sq cm, find AB (in cm).

Introduction / Context:
This question tests the area formula of a trapezium (trapezoid). A trapezium has one pair of parallel sides (here AB and CD). The area is the average of the parallel sides multiplied by the perpendicular distance (height) between them: Area = (1/2) * (AB + CD) * height. We are given CD and height and the area, so we can solve for AB using a simple linear equation. Since AB is stated to be less than CD, the solution should come out less than 6 cm, which provides a useful reasonableness check at the end.

Given Data / Assumptions:

• AB || CD
• CD = 6 cm
• Height (distance between parallel sides) = 4 cm
• Area = 16 sq cm
• Area formula: (1/2) * (AB + CD) * height

Concept / Approach:
Substitute into the trapezium area formula and solve for AB: 16 = (1/2) * (AB + 6) * 4.

Step-by-Step Solution:
Area = (1/2) * (AB + CD) * height 16 = (1/2) * (AB + 6) * 4 (1/2) * 4 = 2, so 16 = 2 * (AB + 6) Divide by 2: 8 = AB + 6 AB = 8 - 6 = 2 cm

Verification / Alternative check:
Plug AB=2 back: area = (1/2)*(2+6)*4 = (1/2)*8*4 = 16 sq cm. It matches exactly, and AB=2 is indeed less than CD=6, consistent with the condition AB < CD.

Why Other Options Are Wrong:
1 cm and 3 cm produce areas 14 or 18 sq cm, not 16. 8 cm violates AB < CD and would also increase area a lot. 4 cm would give area (1/2)*(10)*4 = 20 sq cm.

Common Pitfalls:
Forgetting the 1/2 factor, using CD-AB instead of CD+AB, or using the slant side instead of height as the perpendicular distance.

Final Answer:
AB = 2 cm.