A parallelogram has sides 30 m and 14 m, and one of its diagonals is 40 m long. Using this information, find the area of the parallelogram in square metres (sq m).

Introduction:
This question tests area computation of a parallelogram using sides and a diagonal. A diagonal of a parallelogram divides it into two congruent triangles. If we know the two sides and the diagonal, we effectively know the three sides of one of those triangles. Once we have a triangle with all three sides known, we can compute its area using Heron’s formula. Since the diagonal creates two equal triangles, the area of the parallelogram is twice the area of that triangle.

Given Data / Assumptions:

• Parallelogram sides = 30 m and 14 m• One diagonal = 40 m• Diagonal divides the parallelogram into two congruent triangles

Concept / Approach:
Consider one triangle formed by the diagonal. Its sides are 30 m, 14 m, and 40 m. Use Heron’s formula: if s is semi-perimeter, triangle area = sqrt(s(s-a)(s-b)(s-c)). Then parallelogram area = 2 * triangle area.

Step-by-Step Solution:
Step 1: Identify the triangle sides.Triangle sides: a = 30, b = 14, c = 40Step 2: Compute semi-perimeter.s = (a + b + c)/2 = (30 + 14 + 40)/2 = 84/2 = 42Step 3: Apply Heron’s formula for triangle area.Area_triangle = sqrt(42*(42-30)*(42-14)*(42-40))Area_triangle = sqrt(42*12*28*2)Step 4: Multiply inside step-by-step.42*12 = 50428*2 = 56504*56 = 28224Area_triangle = sqrt(28224) = 168Step 5: Parallelogram area = 2 * 168 = 336

Verification / Alternative check:
A triangle with sides 30, 14, 40 is valid because 30 + 14 > 40 (44 > 40). The computed triangle area 168 is reasonable given the side sizes. Doubling it gives 336 sq m for the parallelogram. Since the diagonal splits the shape into two equal triangles, doubling is correct and ensures internal consistency.

Why Other Options Are Wrong:
136 and 236 are too small to match the triangle-and-double method based on Heron’s formula.436 is too large and would imply a much bigger height than possible with sides 30 and 14.168 is only the area of one triangle, not the full parallelogram.

Common Pitfalls:
• Forgetting to double the triangle area to get parallelogram area.• Using 1/2 * base * height without actually finding the height.• Arithmetic mistakes inside Heron’s formula multiplication.

Final Answer:
The area of the parallelogram is 336 sq m.