A parallelogram has sides 30 m and 14 m and one of its diagonals is 40 m long. Then, its area is

### Concept & Logic A diagonal divides a parallelogram into two congruent triangles. By finding the area of one of these triangles using Heron's formula and doubling it, we can find the total area of the parallelogram. $$\text{Heron's Formula: } Area = \sqrt{s(s-a)(s-b)(s-c)}$$ where $s$ is the semi-perimeter: $s = \frac{a+b+c}{2}$ ### Step-by-Step Solution 1. **Identify the triangle sides:** * The diagonal and the two adjacent sides of the parallelogram form a triangle with sides $a = 30$, $b = 14$, and $c = 40$. 2. **Calculate the semi-perimeter ($s$):** * $s = \frac{30 + 14 + 40}{2} = \frac{84}{2} = 42 \text{ m}$ 3. **Calculate the area of one triangle:** * Area $= \sqrt{42 \times (42 - 30) \times (42 - 14) \times (42 - 40)}$ * Area $= \sqrt{42 \times 12 \times 28 \times 2}$ * Prime factorization helps simplify: $\sqrt{(14 \times 3) \times (4 \times 3) \times (14 \times 2) \times 2}$ * Area $= \sqrt{14^2 \times 3^2 \times 4 \times 2^2} = 14 \times 3 \times 2 \times 2 = 168 \text{ m}^2$ 4. **Calculate total area:** * Total Area $= 2 \times 168 = 336 \text{ m}^2$ ### Exam Strategy & Shortcut When using Heron's formula with large numbers, never multiply them out into a massive product! Break them into smaller factors (especially pairs like two 14s, two 3s) and pull them directly out of the square root. ### Common Pitfall Calculating the area of just the one triangle ($168 \text{ m}^2$) and forgetting to multiply by 2 for the full parallelogram. (Notice 168 is conveniently listed as option A). ### Final Answer Therefore, the correct answer is **336 m²**.