The two parallel sides of a trapezium measure 21 cm and 9 cm, and its height is 10 cm. What is the area of the trapezium in square centimetres?

Introduction / Context:
This question tests knowledge of the area formula for a trapezium, a standard topic in mensuration and aptitude geometry. A trapezium has one pair of parallel sides, often called the bases. When the lengths of these bases and the height are known, the area can be computed directly using a simple formula that averages the lengths of the parallel sides and multiplies by the height. This type of question focuses on correct formula recall and careful substitution of values.

Given Data / Assumptions:
- The figure is a trapezium with exactly one pair of parallel sides. - Length of the first parallel side (base one) is 21 cm. - Length of the second parallel side (base two) is 9 cm. - The perpendicular distance between the parallel sides, that is the height, is 10 cm. - We need the area in square centimetres.

Concept / Approach:
The standard area formula for a trapezium is Area = (1 / 2) * (sum of parallel sides) * height. This can be viewed as the average of the two parallel side lengths multiplied by the height. Since all measurements are already in centimetres, there is no need for unit conversion. The question is a direct formula application, so the main focus should be on adding the bases correctly and doing the multiplication carefully.

Step-by-Step Solution:
Step 1: Let the lengths of the parallel sides be a = 21 cm and b = 9 cm. Step 2: Let the height of the trapezium be h = 10 cm. Step 3: Use the formula Area = (1 / 2) * (a + b) * h. Step 4: Compute the sum of the parallel sides: a + b = 21 + 9 = 30. Step 5: Substitute into the formula: Area = (1 / 2) * 30 * 10. Step 6: Simplify (1 / 2) * 30 = 15. Step 7: Multiply 15 by 10 to get Area = 150 square centimetres.

Verification / Alternative check:
An alternative way to think about the formula is that a trapezium can be decomposed into a rectangle and two right triangles, but this is more complex here. The result 150 also respects the fact that if both parallel sides were 21 cm, the area would be 21 * 10 = 210. Since one side is smaller, the actual area must be less than 210. Similarly, if both sides were 9 cm, the area would be 9 * 10 = 90, so the true area must be between 90 and 210. The value 150 lies between these bounds, which supports the computation.

Why Other Options Are Wrong:
Option 35 is far too small and does not follow from the area formula for the given dimensions. Option 75 is exactly half of the correct area and would correspond to using only one of the parallel sides instead of their average. Option 225 is larger than the area of a rectangle of size 21 cm by 10 cm, so it cannot be correct for this trapezium where one base is shorter.

Common Pitfalls:
Learners sometimes confuse the formula for the area of a trapezium with that of a rectangle or a triangle. Another mistake is adding the height to the parallel sides instead of multiplying by it. Forgetting the factor of one half is also common and usually leads to exactly double the correct result. Keeping the formula Area = (1 / 2) * (sum of parallel sides) * height clearly in mind helps avoid these errors.

Final Answer:
The area of the trapezium is 150 square centimetres.