The area of a trapezium is 21 square centimetres, the distance between its two parallel sides is 3 cm, and one of the parallel sides is 8.3 cm. What is the length of the other parallel side?

Introduction / Context:
This problem deals with the area of a trapezium (trapezoid), which has two parallel sides called bases and a distance between them called height. You are given the area, the height (the distance between the parallel sides), and one base, and you must find the other base. Questions of this type are very typical in geometry-based aptitude tests and help reinforce the area formula for trapeziums.

Given Data / Assumptions:

• Area of the trapezium A = 21 square centimetres.
• Distance between the parallel sides (height) h = 3 cm.
• One of the parallel sides, say a, is 8.3 cm.
• The other parallel side, say b, is unknown.
• We use the standard area formula for a trapezium.

Concept / Approach:
The area A of a trapezium with parallel sides a and b and height h is given by:
A = (1 / 2) * (a + b) * h. We are given A, h, and a, and we must solve for b. Rearranging the formula allows us to isolate a + b and then find b by subtracting a.

Step-by-Step Solution:
Step 1: Write the area formula: A = (1 / 2) * (a + b) * h. Step 2: Substitute A = 21, h = 3, and a = 8.3: 21 = (1 / 2) * (8.3 + b) * 3. Step 3: Simplify the factor (1 / 2) * 3 = 3 / 2. Step 4: So 21 = (3 / 2) * (8.3 + b). Step 5: Multiply both sides by 2 / 3 to clear the fraction: 8.3 + b = (21 * 2 / 3). Step 6: Compute (21 * 2 / 3) = 14. Step 7: So 8.3 + b = 14, hence b = 14 - 8.3 = 5.7 cm. Step 8: Therefore, the other parallel side has length 5.7 cm.

Verification / Alternative check:
Check by plugging back into the area formula. Compute (a + b) = 8.3 + 5.7 = 14. Then area A = (1 / 2) * 14 * 3 = 7 * 3 = 21 square centimetres, which matches the given area exactly. This confirms that the calculated length b = 5.7 cm is correct.

Why Other Options Are Wrong:
If the other side were 12.7 cm, then a + b would be 21.0, giving area (1 / 2) * 21 * 3 = 31.5, which is too large. If b were 7.7 cm, then a + b = 16, giving area (1 / 2) * 16 * 3 = 24, again too large. For 10.7 cm, the sum is 19, giving area 28.5. A value of 9.7 cm also gives an incorrect area. Only b = 5.7 cm restores the area to 21 square centimetres.

Common Pitfalls:
A common mistake is to forget the factor of 1 / 2 in the area formula or to treat 8.3 as an approximate 8, which produces incorrect arithmetic. Another error is solving for a + b correctly but then mis-subtracting the known side, leading to a wrong length. Careful handling of decimal subtraction (14 - 8.3) is important to avoid minor but crucial numerical errors.

Final Answer:
The length of the other parallel side of the trapezium is 5.7 cm.