In a trapezium ABCD, sides AB and CD are parallel, and ∠ADC = 90°. If AB = 15 cm, CD = 40 cm and the diagonal AC = 41 cm, what is the area of trapezium ABCD in cm²?

Introduction:
This geometry question deals with a right-angled trapezium, where one non-parallel side is perpendicular to the parallel sides. You are given the lengths of the two parallel sides and a diagonal, and asked to find the area of the trapezium. The key is to interpret the trapezium as composed of right triangles and a rectangle or to use coordinates to find the height.

Given Data / Assumptions:

• AB ∥ CD in trapezium ABCD.
• ∠ADC = 90°.
• AB = 15 cm (shorter parallel side).
• CD = 40 cm (longer parallel side).
• Diagonal AC = 41 cm.
• We must find area(ABCD) in cm².

Concept / Approach:
Place the trapezium on a coordinate plane to find its height. Since ∠ADC = 90° and CD is parallel to AB, we can take CD as horizontal and AD as vertical. The diagonal AC length will then allow us to compute the vertical height of the trapezium. Finally, use the area formula for a trapezium: Area = (1/2) * (sum of parallel sides) * height.

Step-by-Step Solution:
Let D = (0, 0), C = (40, 0). Then CD = 40 cm on the x-axis.Since ∠ADC = 90°, AD is perpendicular to CD, so AD is vertical. Let A = (0, h).AB is parallel to CD, so AB is also horizontal at y = h. Since AB = 15 cm, B = (15, h).Diagonal AC connects A(0, h) to C(40, 0) and has length 41 cm.Use distance formula for AC:AC² = (40 − 0)² + (0 − h)² = 40² + h² = 1600 + h².Given AC = 41, so AC² = 41² = 1681.Thus 1600 + h² = 1681 ⇒ h² = 81 ⇒ h = 9 cm (height of trapezium).Area of trapezium = (1/2) * (AB + CD) * h = (1/2) * (15 + 40) * 9.Sum of parallel sides = 55, so area = (1/2) * 55 * 9 = 27.5 * 9 = 247.5 cm².

Verification / Alternative check:
You can view the trapezium as a large right triangle plus or minus a smaller right triangle, but all consistent constructions give height 9 cm and the same area. The use of Pythagoras with diagonal 41 and base 40 naturally produces a 9-40-41 triangle, which is a known Pythagorean triple, giving confidence in the height value.

Why Other Options Are Wrong:
Values 245, 240, 250 and 260 cm² correspond to incorrect assumed heights or arithmetic mistakes (for example, using the wrong base sum or miscomputing the Pythagorean relationship). Only 247.5 cm² is consistent with the height of 9 cm and the given sides.

Common Pitfalls:
Errors often occur when interpreting the right angle position, incorrectly assigning coordinates, or using AB or CD as the height instead of a perpendicular distance. Another common mistake is to forget that area of a trapezium uses the average of the parallel sides times the height, not simply base times height.

Final Answer:
The area of trapezium ABCD is 247.5 cm².