Problem 8 - Trigonometry

Question

Find the semi perimeter of triangle \(A B C\). \(a=153 \mathrm{~cm}, b=174 \mathrm{~cm}, c=232 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The semi-perimeter of triangle \(ABC\) is 279.5 cm.

Step 1: Understanding the Semi-Perimeter Concept

The semi-perimeter of a triangle is defined as half of the sum of all its sides. If a triangle has sides of lengths \(a\), \(b\), and \(c\), its semi-perimeter \(s\) can be calculated using the formula \(s = \frac{a+b+c}{2}\).

Step 2: Using Given Side Lengths

Substitute the provided side lengths \(a = 153\) cm, \(b = 174\) cm, and \(c = 232\) cm into the semi-perimeter formula: \[s = \frac{153 + 174 + 232}{2}\]

Step 3: Calculating the Total Perimeter

First, find the sum of all the side lengths: \(153 + 174 + 232\). Calculate this sum to find the total perimeter of the triangle.

Step 4: Finding the Sum

Perform the addition: \(153 + 174 + 232 = 559\). This is the total perimeter of triangle \(ABC\).

Step 5: Calculating the Semi-Perimeter

Now, divide the total perimeter by 2 to find the semi-perimeter: \[s = \frac{559}{2}\]

Step 6: Final Calculation of Semi-Perimeter

Perform the division to calculate the semi-perimeter: \(s = 279.5\) cm.

Key Concepts

Understanding Triangle Side Lengths

A triangle is a three-sided polygon, and its side lengths are essential in many calculations. In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem. For triangle \(ABC\) with side lengths \(a\), \(b\), and \(c\), this theorem ensures stability and a valid geometric shape.

In our problem, the side lengths are explicitly given as \(a = 153\) cm, \(b = 174\) cm, and \(c = 232\) cm. These values are necessary inputs for calculating not only perimeter but also for other geometrical properties, such as area and semi-perimeter.

It's important to confirm that the addition of side lengths follows the inequality:

• \(a + b > c\)
• \(a + c > b\)
• \(b + c > a\)

These conditions help in verifying that the given side lengths form a valid triangle.

Perimeter Calculation of a Triangle

The perimeter of a triangle is the total distance around the triangle, calculated by summing up its side lengths. For triangle \(ABC\), the perimeter \(P\) is given by the formula:
\[P = a + b + c\]
Using the side lengths from our problem, we substitute the values:
\[P = 153 + 174 + 232\]
Adding these values together, we find that the perimeter is 559 cm. This calculation provides a straightforward way of understanding how much material would be needed to enclose the triangle fully, such as estimating fencing for a triangular garden.
Accurate perimeter calculation plays a significant role in further computations, like finding the semi-perimeter, which simplifies other complex calculations related to the triangle.

Semi-Perimeter Formula and Calculation

The semi-perimeter is a useful geometric property of a triangle. It is half of the perimeter and is particularly important for calculations involving the triangle's area using Heron's formula.
To find the semi-perimeter \(s\), we use the formula:
\[s = \frac{a+b+c}{2}\]
After calculating the perimeter as 559 cm, the semi-perimeter is obtained by dividing this total by 2:
\[s = \frac{559}{2} = 279.5\text{ cm}\]
The semi-perimeter is an excellent midpoint value that simplifies several triangle-related equations, especially when working with equations requiring the right balance between perimeter and area computations. It serves as a pivotal tool in geometry, simplifying more complex mathematical tasks.