Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Heron's Area Formula to find the area of the triangle. $$a=1.24, \quad b=2.45, \quad c=1.25$$

Step-by-Step Solution

Answer

The area of the triangle calculated using Heron's formula is \(\sqrt{s(s-a)(s-b)(s-c)}\) units

Step 1: Calculate the semi-perimeter

The semi-perimeter of the triangle is calculated using the formula: \(s = (a + b + c) / 2\). Plug in the values of \(a\), \(b\), and \(c\) into the formula, substituting 1.24, 2.45, and 1.25 respectively.

Step 2: Compute the area using Heron's formula

The area of the triangle can then be found using Heron's formula: \(Area = \sqrt{s(s-a)(s-b)(s-c)}\). Subsitute the values for \(s\), \(a\), \(b\), and \(c\) into this formula

Step 3: Simplify the expression to get the final result

Simplify the expression obtained in Step 2 to get the result for the area of the triangle

Key Concepts

Triangle Area Calculation

Calculating the area of a triangle is a fundamental concept in geometry, critical for students in understanding spatial relationships and properties of shapes. While there are several methods to calculate the area of a triangle, one approach relies on knowing the lengths of all three sides, without needing to find the height. This method is particularly useful when dealing with irregular triangles, where the height is not easily measurable.

Different formulas can be applied depending on the information available about the triangle. The most basic formula, \( \text{Area} = \frac{1}{2} \cdot base \cdot height \), is applicable when the base and height of the triangle are known. However, when only the side lengths are known, Heron's Area Formula becomes invaluable. It provides a way to compute the area based on the lengths of the sides alone, which is the focus of this article.

Semi-perimeter of a Triangle

The concept of the semi-perimeter of a triangle is essential in understanding and applying Heron's formula effectively. It is defined as half of the perimeter of the triangle, the latter being the sum of the lengths of its three sides. Mathematically, it is expressed as \( s = \frac{a + b + c}{2} \), where \( s \) is the semi-perimeter, and \( a \) , \( b \) , and \( c \) are the lengths of the sides of the triangle.

To further simplify this concept:

• The perimeter is the total distance around the triangle, obtained by adding up all side lengths.
• The semi-perimeter is exactly half of this total distance, serving as a crucial step in solving Heron's formula.
• Understanding the semi-perimeter is not just crucial for Heron's formula, but is also useful in various other areas of mathematics including solving for circumradius, inscribed circle radius, and during calculations in trigonometry.

Solving Heron's Formula

Solving Heron's formula may seem daunting initially, but it becomes straightforward when approached methodically. Heron's formula states that the area of a triangle with sides \( a \), \( b \), and \( c \) is given by \( \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} \), where \( s \) is the semi-perimeter calculated as previously described.

The steps for using Heron's formula are:

• First, calculate the semi-perimeter \( s \) using the side lengths of the triangle.
• Next, plug in the values of the sides \( a \) , \( b \) , \( c \) and the semi-perimeter \( s \) into the formula.
• Carry out the multiplications within the square root carefully.
• Finally, find the square root of the result to obtain the area.

This process requires care during multiplication to avoid calculation errors and to simplify as much as possible before taking the square root for a clean final result. The exercise at hand involves a triangle with sides of 1.24, 2.45, and 1.25 units, requiring these values to be substituted respectively into Heron's formula following the steps above to find the area.