Find the area, to the nearest cm$^2$, of the triangle whose sides are in the ratio 2 : 3 : 4 and whose perimeter is 180 cm.

1. Analyze the question

We are given the ratio of the sides of a triangle (2:3:4) and its perimeter (180 cm). We need to find the area of this triangle using Heron's formula, rounding the result to the nearest whole number.

2. Explain the concept

• Finding the sides: When sides are in a ratio, we represent them as $2x$, $3x$, and $4x$. The perimeter is the sum of these sides.
• Heron's Formula: To find the area of a triangle when all three sides ($a, b, c$) are known, we use: $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$where$s$is the semi-perimeter:$s = \frac{a+b+c}{2}$.

3. Show the solution

Step 1: Find the actual lengths of the sides. The perimeter is the sum of the sides: $2x + 3x + 4x = 180$ $9x = 180$ $x = \frac{180}{9} = 20$

So, the sides are: $a = 2(20) = 40 \text{ cm}$ $b = 3(20) = 60 \text{ cm}$ $c = 4(20) = 80 \text{ cm}$

Step 2: Calculate the semi-perimeter ($s$). $s = \frac{\text{Perimeter}}{2} = \frac{180}{2} = 90 \text{ cm}$

Step 3: Apply Heron's Formula. $\text{Area} = \sqrt{90(90-40)(90-60)(90-80)}$ $\text{Area} = \sqrt{90 \times 50 \times 30 \times 10}$ $\text{Area} = \sqrt{1,350,000}$

Step 4: Calculate the square root. $\sqrt{1,350,000} \approx 1161.895$

Rounding to the nearest cm$^2$, we get 1162 cm$^2$.

4. Identify the answer

Comparing our result to the options provided, 1162 cm$^2$ corresponds to option A.

5. Tip

When dealing with ratios in geometry, always find the value of the "part" ($x$) first. Also, for Heron's formula, if the numbers are large, look for factors that are perfect squares to simplify the square root calculation without needing a calculator!