Two adjacent sides of a parallelogram measure 21 cm and 20 cm. The diagonal joining the endpoints of these two sides is 29 cm. What is the area of the parallelogram (in square centimetres)?

Introduction / Context:
This question combines properties of parallelograms with the Pythagoras theorem. A parallelogram has opposite sides parallel and equal, and its area is given by the product of the base and the corresponding height. However, when the height is not directly provided, you can sometimes use diagonals and side lengths to deduce special cases. Here, the lengths of two adjacent sides and the diagonal between them are given, which allows you to check whether the parallelogram is in fact a rectangle. Once you recognise this, finding the area becomes straightforward.

Given Data / Assumptions:
• The figure is a parallelogram with adjacent sides of lengths 21 cm and 20 cm.
• The diagonal joining the endpoints of these two sides has length 29 cm.
• The parallelogram lies in a Euclidean plane and has positive area.
• We are asked to find the area of the parallelogram in square centimetres.

Concept / Approach:
Consider the triangle formed by the two adjacent sides and the included diagonal. Let these sides be a = 21 cm and b = 20 cm, and the diagonal be d = 29 cm. In a general parallelogram, the diagonal satisfies the law of cosines: d^2 = a^2 + b^2 - 2ab cos(θ), where θ is the angle between the sides. If this triangle turns out to be a right triangle, then cos(θ) = 0, which means θ = 90°. That would indicate that the parallelogram is actually a rectangle, and its area is simply a * b. So we test whether 21, 20, and 29 form a Pythagorean triple.

Step-by-Step Solution:
Step 1: Compute a^2 + b^2 for the given side lengths: 21^2 + 20^2. Step 2: Calculate: 21^2 = 441 and 20^2 = 400, so a^2 + b^2 = 441 + 400 = 841. Step 3: Compute d^2 for the diagonal: 29^2 = 841. Step 4: Since a^2 + b^2 = d^2, the triangle formed by sides 21 cm, 20 cm and diagonal 29 cm is a right triangle, and θ = 90°. Step 5: Therefore, the parallelogram is a rectangle with side lengths 21 cm and 20 cm. Step 6: The area of a rectangle is side1 * side2 = 21 * 20 = 420 square centimetres.

Verification / Alternative check:
The fact that 21^2 + 20^2 equals 29^2 exactly confirms the right angle between the sides. For a parallelogram with sides a and b and included angle θ, the area is a * b * sin(θ). Here, sin(90°) = 1, so the area reduces to a * b = 21 * 20 = 420, matching our calculation. There is no need for further adjustments or height calculations because the right angle already provides the maximum area for given side lengths.

Why Other Options Are Wrong:
Option 240 square centimetres would correspond to a smaller height than 20 cm for a base of 21 cm and does not reflect the right angle condition. Option 120 square centimetres is far too small and ignores the actual geometry of the sides and diagonal. Option 210 square centimetres is half of the correct result and may arise if someone mistakenly computes the area of the right triangle instead of the full parallelogram. Only 420 square centimetres correctly interprets the shape as a rectangle and uses the correct base and height.

Common Pitfalls:
Some students fail to check whether the parallelogram is a rectangle and instead try to guess the height or use incorrect formulas. Others may mistakenly think that the diagonal plays a direct role in the area formula. The key is to notice that the given side lengths and diagonal form a Pythagorean triple, signalling a right angle between the sides. Once that is recognised, the problem becomes very simple. Always verify whether a given triple satisfies a^2 + b^2 = c^2, as this often simplifies geometry questions dramatically.

Final Answer:
The area of the parallelogram is 420 square centimetres.