Triangle Area Calculation Given Semi-Perimeter And Side Ratios
This article will walk you through the process of calculating the area of a triangle when you are given its semi-perimeter and the ratio of its sides. This is a classic geometry problem that combines the concepts of ratios, perimeter, and Heron's formula. We will break down each step in detail to ensure a clear understanding of the solution. Let’s dive in and explore how to solve this problem systematically.
Understanding the Problem
Before we start crunching numbers, let's make sure we understand what the problem is asking. We are given a triangle with a semi-perimeter of 96 cm. The sides of the triangle are in the ratio 3:4:5. Our goal is to find the area of this triangle. To do this effectively, we need to recall some key concepts and formulas. First, let's define the semi-perimeter. The semi-perimeter (s) of a triangle is half the sum of the lengths of its sides. If the sides are a, b, and c, then:
s = (a + b + c) / 2
Next, we need to remember Heron's formula, which allows us to calculate the area of a triangle when we know the lengths of all three sides. Heron's formula states that the area (A) of a triangle is given by:
A = √[s(s - a)(s - b)(s - c)]
Where s is the semi-perimeter, and a, b, and c are the lengths of the sides. Now that we have the necessary formulas, let's apply them to the given problem. The sides are in the ratio 3:4:5, which means we can represent the sides as 3x, 4x, and 5x, where x is a common ratio. Our semi-perimeter is 96 cm, which means we can set up an equation to find the value of x. Once we find x, we can determine the actual lengths of the sides and then use Heron's formula to calculate the area.
Step-by-Step Solution
1. Expressing Sides in Terms of a Ratio
The problem states that the sides of the triangle are in the ratio 3:4:5. This means that the lengths of the sides can be represented as 3x, 4x, and 5x, where 'x' is a common multiplier. This representation maintains the proportion while allowing us to work with concrete lengths. For example, if x were 10, the sides would be 30 cm, 40 cm, and 50 cm. The key here is that we are not yet sure what 'x' is, but we have a way to express the sides relative to each other.
2. Using the Semi-Perimeter to Find 'x'
We know that the semi-perimeter (s) is half the sum of the sides. Given that the semi-perimeter is 96 cm, we can write the equation:
s = (3x + 4x + 5x) / 2
Substituting the given value of the semi-perimeter:
96 = (3x + 4x + 5x) / 2
Combine the terms on the right side:
96 = (12x) / 2
96 = 6x
Now, solve for 'x':
x = 96 / 6
x = 16
So, the value of x is 16. This is a crucial step because it allows us to find the actual lengths of the sides of the triangle. Without knowing x, we could only express the sides in terms of a ratio, but now we can find their exact measurements.
3. Calculating the Side Lengths
Now that we have found x = 16, we can calculate the lengths of the sides of the triangle:
- Side a = 3x = 3 * 16 = 48 cm
- Side b = 4x = 4 * 16 = 64 cm
- Side c = 5x = 5 * 16 = 80 cm
We now know that the sides of the triangle are 48 cm, 64 cm, and 80 cm. These are the values we need to use in Heron's formula to find the area of the triangle. Before we apply Heron's formula, it's a good idea to double-check our semi-perimeter calculation using these side lengths:
s = (48 + 64 + 80) / 2
s = 192 / 2
s = 96 cm
This confirms that our calculations are correct, and we can proceed with Heron's formula.
4. Applying Heron's Formula
Heron's formula states that the area (A) of a triangle is given by:
A = √[s(s - a)(s - b)(s - c)]
Where s is the semi-perimeter, and a, b, and c are the lengths of the sides. We have already found that s = 96 cm, a = 48 cm, b = 64 cm, and c = 80 cm. Plugging these values into Heron's formula:
A = √[96(96 - 48)(96 - 64)(96 - 80)]
A = √[96 * 48 * 32 * 16]
Now, let's break down the calculation:
A = √[96 * 48 * 32 * 16]
A = √[(3 * 32) * (3 * 16) * 32 * 16]
A = √[3 * 3 * 32 * 32 * 16 * 16]
A = 3 * 32 * 16
A = 1536
Therefore, the area of the triangle is 1536 cm². This is the final answer, and it matches one of the given options. The step-by-step application of Heron's formula, along with the correct side lengths and semi-perimeter, leads us to the accurate solution.
Final Answer
The area of the triangle is 1536 cm². So, the correct option is B. This problem illustrates how a combination of basic geometry principles and algebraic manipulation can be used to solve complex problems. Understanding the ratios, semi-perimeter, and Heron's formula is essential for tackling such questions efficiently and accurately. By breaking the problem down into smaller, manageable steps, we can navigate through the calculations and arrive at the correct answer.
