ABCD is a quadrilateral with AB = 9 cm, BC = 40 cm, CD = 28 cm, DA = 15 cm and angle ∠ABC is a right angle. What is the area of quadrilateral ABCD in square centimetres?

Introduction / Context:
This question involves finding the area of a quadrilateral by decomposing it into two triangles. The quadrilateral has one right angle at B and four given side lengths. By drawing the diagonal from A to C, we create a right triangle at B and a second triangle with all three sides known. The area of the right triangle can be computed using the simple half base times height formula, while the area of the other triangle can be found using Heron's formula. Adding the two areas gives the area of the quadrilateral.

Given Data / Assumptions:
- Quadrilateral ABCD has sides AB = 9 cm, BC = 40 cm, CD = 28 cm and DA = 15 cm. - Angle ∠ABC is a right angle, so AB ⟂ BC. - Diagonal AC divides ABCD into triangles ABC and ACD. - All sides and angles are in a plane and form a valid quadrilateral.

Concept / Approach:
We first focus on right triangle ABC. With AB and BC perpendicular, its area is straightforward: (1 / 2) * AB * BC. The diagonal AC serves as the hypotenuse, which can be found using the Pythagoras theorem. Then triangle ACD has sides AC, CD and AD all known, so Heron's formula provides its area. Heron's formula uses the semi perimeter and the three side lengths. The total area of quadrilateral ABCD is then the sum of the areas of triangles ABC and ACD.

Step-by-Step Solution:
Step 1: Consider triangle ABC, which is right angled at B. Step 2: Its legs are AB = 9 cm and BC = 40 cm. Step 3: Area of triangle ABC is (1 / 2) * AB * BC = (1 / 2) * 9 * 40 = 180 cm². Step 4: Find the diagonal AC using Pythagoras theorem: AC² = AB² + BC² = 9² + 40². Step 5: Compute 9² = 81 and 40² = 1600, so AC² = 81 + 1600 = 1681. Step 6: Taking the square root gives AC = 41 cm. Step 7: Now consider triangle ACD with sides AC = 41 cm, CD = 28 cm and AD = 15 cm. Step 8: Compute its semi perimeter s = (41 + 28 + 15) / 2 = 84 / 2 = 42 cm. Step 9: Apply Heron's formula: Area = √[s(s − a)(s − b)(s − c)] where a, b, c are 41, 28 and 15. Step 10: Compute the factors: s − a = 42 − 41 = 1, s − b = 42 − 28 = 14, s − c = 42 − 15 = 27. Step 11: Multiply inside the root: 42 * 1 * 14 * 27 = 42 * 14 * 27. Step 12: Evaluate 42 * 14 = 588 and 588 * 27 = 15876. Step 13: The area of triangle ACD is √15876, which simplifies to 126 cm² because 126² = 15876. Step 14: Total area of quadrilateral ABCD = area(ABC) + area(ACD) = 180 + 126 = 306 cm².

Verification / Alternative check:
We can verify that sides 41, 28 and 15 can form a valid triangle by checking the triangle inequality: 28 + 15 = 43 > 41, 41 + 15 > 28 and 41 + 28 > 15. All inequalities hold, so triangle ACD is valid. The right triangle ABC is also valid by construction. The exact square of 126 is 15876, confirming that the Heron's formula evaluation is correct. The sum 180 + 126 equals 306, which must be one of the provided options and matches a reasonable area size for the given side lengths.

Why Other Options Are Wrong:
Option 300 would be obtained if there were small numerical errors in the Heron computation or rounding, but it does not match the exact square root result. Option 312 and Option 316 are larger than 306 and would require either a longer diagonal or larger side lengths than those given.

Common Pitfalls:
Common mistakes include miscomputing the diagonal AC, forgetting to use the correct legs for the right triangle area, or making arithmetic errors when multiplying the Heron factors. Some students also incorrectly use 28 or 15 as a height for the quadrilateral instead of decomposing it into triangles. Drawing a clear diagram with diagonal AC and carefully applying Pythagoras and Heron's formula step by step is the best way to avoid these issues.

Final Answer:
The area of quadrilateral ABCD is 306 square centimetres.