The height of a triangle is equal to the perimeter of a square whose diagonal is $8\sqrt{2}$ metre and the base of the same triangle is equal to the side of a square whose area is 729sq. metre. What is the area of the triangle? (In sq. metre) [United India Insurance (UIICL) Assistant (Online) Exam, 2015]

### Concept & Multi-Shape Integration This problem connects the properties of squares (diagonal, perimeter, area, side) to the dimensions of a triangle (base, height) to calculate the triangle's area. $$Diagonal_{square} = a\sqrt{2}$$ $$Area_{square} = s^2$$ $$Area_{triangle} = \frac{1}{2} \times base \times height$$ ### Step-by-Step Solution * **Step 1: Find the triangle's height.** * The diagonal of the first square is $8\sqrt{2}$ m. * Formula for diagonal is $side \times \sqrt{2}$. Thus, the side of the first square is 8 m. * The perimeter of this square is $4 \times 8 = 32$ m. * Given: Height of triangle = Perimeter of this square = 32 m. * **Step 2: Find the triangle's base.** * The area of the second square is 729 sq. m. * Formula for area is $side^2 = 729$. Taking the square root, the side is $\sqrt{729} = 27$ m. * Given: Base of triangle = Side of this square = 27 m. * **Step 3: Calculate the area of the triangle.** * Area = $\frac{1}{2} \times base \times height$. * Area = $\frac{1}{2} \times 27 \times 32$. * Area = $27 \times 16$. * $27 \times 16 = 27 \times (10 + 6) = 270 + 162 = 432$ sq. m. ### Exam Strategy & Shortcut Since you need to multiply $\frac{1}{2} \times 27 \times 32$, you can just calculate the unit digit. The unit digit of $27 \times 16$ is $7 \times 6 = 42$, which ends in 2. Looking at the options, both (b) 206 and (c) 472 and (d) 432 end in 2, but $27 \times 16$ is close to $25 \times 16 = 400$, making 432 the only logical choice. ### Common Pitfall Confusing the area of the square with its side or perimeter, or forgetting the $\frac{1}{2}$ multiplier in the triangle area formula. ### Final Answer Therefore, the correct answer is **432**.