Problem 9 - Trigonometry

Question

Find the semi perimeter of triangle \(A B C\). \(a=2.1 \mathrm{~m}, b=2.3 \mathrm{~m}, c=3.9 \mathrm{~m}\)

Step-by-Step Solution

Answer

The semi-perimeter of triangle \(A B C\) is \(4.15\, m\).

Step 1: Understand the Semi-Perimeter Formula

The semi-perimeter of a triangle is defined as half of the triangle's perimeter. The formula for calculating the semi-perimeter, usually denoted as \(s\), is: \[ s = \frac{a + b + c}{2} \] where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.

Step 2: Substitute the Given Values

Using the given side lengths of triangle \(A B C\), which are \(a = 2.1\, m\), \(b = 2.3\, m\), and \(c = 3.9\, m\), substitute these values into the semi-perimeter formula: \[ s = \frac{2.1 + 2.3 + 3.9}{2} \]

Step 3: Calculate the Sum of the Sides

Add the lengths of the sides together: \(2.1 + 2.3 + 3.9 = 8.3\, m\). This is the total perimeter of the triangle.

Step 4: Divide by 2 to Get the Semi-Perimeter

Using the sum of side lengths, divide by 2 to calculate the semi-perimeter: \[ s = \frac{8.3}{2} = 4.15\, m \]

Step 5: Conclusion of Calculations

The semi-perimeter of triangle \(A B C\) is \(4.15\, m\).

Key Concepts

Understanding the Perimeter of a Triangle

A triangle's perimeter is the total distance around the triangle, which means it is the sum of the lengths of all its sides. To find the perimeter of a triangle, you simply add up the lengths of its three sides. It is generally represented as:

• Perimeter = side1 + side2 + side3

For our triangle ABC, with sides measuring:

• \(a = 2.1 \text{ m}\)
• \(b = 2.3 \text{ m}\)
• \(c = 3.9 \text{ m}\)

The perimeter would be the sum of these, i.e., \(2.1 + 2.3 + 3.9 = 8.3 \text{ m}\). Understanding this step is important as it feeds directly into finding the semi-perimeter.

Exploring Triangle Side Lengths

The side lengths of a triangle determine its shape and size. In a triangle, knowing the lengths of all three sides enables you to calculate many other properties, such as the perimeter, semi-perimeter, and even the area using more advanced formulas like Heron's formula.

Each side length plays a crucial role:

• The longest side of a right triangle is the hypotenuse.
• In an equilateral triangle, all sides are equal, simplifying calculations.
• In our triangle, the side lengths do not follow these specific patterns, as it's neither isosceles nor equilateral, but knowing they are \(a = 2.1 \text{ m}\), \(b = 2.3 \text{ m}\), \(c = 3.9 \text{ m}\) helps us to define its form and calculate important values such as the semi-perimeter efficiently.

Understanding the Semi-Perimeter Formula

The semi-perimeter formula is a handy tool often used in geometry, especially concerning triangle calculations. The semi-perimeter, denoted by \(s\), is simply half of the triangle's perimeter. It's useful in finding other things, like the area of a triangle using the Heron's formula. The formula is:

• \(s = \frac{a + b + c}{2}\)

Here, \(a\), \(b\), and \(c\) are the side lengths of the triangle.

For triangle ABC, substituting the given side lengths:

• \(s = \frac{2.1 + 2.3 + 3.9}{2}\)

Which simplifies to \(s = \frac{8.3}{2} = 4.15 \text{ m}\). This calculation gives us the semi-perimeter, a critical value applied in further explorations of triangle properties.