Problem 7 - Trigonometry

Question

Find the semi perimeter of triangle \(A B C\). \(a=3 \mathrm{ft}, b=4 \mathrm{ft}, c=5 \mathrm{ft}\)

Step-by-Step Solution

Answer

The semi-perimeter of triangle \( ABC \) is 6 ft.

Step 1: Understand the Problem

We need to find the semi-perimeter of triangle \( ABC \) given side lengths \( a = 3 \) ft, \( b = 4 \) ft, and \( c = 5 \) ft. The semi-perimeter is half of the triangle's perimeter.

Step 2: Find the Perimeter

To find the semi-perimeter, we first calculate the perimeter of the triangle. The formula for the perimeter \( P \) of a triangle is the sum of its side lengths: \[ P = a + b + c \] Substitute the given side lengths: \[ P = 3 + 4 + 5 \] \[ P = 12 \text{ ft} \]

Step 3: Calculate the Semi-Perimeter

The semi-perimeter \( s \) is half of the perimeter. Thus, the formula for the semi-perimeter is: \[ s = \frac{P}{2} \] Substitute the perimeter value: \[ s = \frac{12}{2} \] \[ s = 6 \text{ ft} \]

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is the total length around the triangle. It is calculated by adding together the lengths of all three sides. If you imagine walking along the edges of a triangle, the distance you would cover in total is its perimeter. For any given triangle with sides labeled as \( a \), \( b \), and \( c \), the formula to find its perimeter, \( P \), is:\[P = a + b + c\]In our exercise, we have a triangle with sides 3 ft, 4 ft, and 5 ft. Adding these values:

• 3 ft + 4 ft + 5 ft = 12 ft

So, the perimeter of triangle ABC is 12 feet. Knowing the perimeter is crucial because it helps us find other properties like the semi-perimeter.

Understanding Triangle Geometry

Triangles are fascinating geometric shapes and are a basic building block for more complex figures. In geometry, a triangle consists of three sides, three vertices (corners), and three angles. Triangles can be classified based on their sides or angles:

• **By Sides:** Equilateral (all sides equal), Isosceles (two sides equal), Scalene (all sides different).
• **By Angles:** Acute (all angles < 90°), Right (one angle = 90°), Obtuse (one angle > 90°).

For our triangle ABC, since the side lengths are 3 ft, 4 ft, and 5 ft, each side is different, making it a scalene triangle. Furthermore, because it satisfies the Pythagorean theorem \((3^2 + 4^2 = 5^2)\), it's also a right triangle. Understanding the type of triangle you are working with is essential when solving various geometrical problems.

Calculating Triangle Sides

Calculating triangle sides might seem straightforward when you have all the numbers, but there are important principles involved. Often, you use the side lengths to find other properties such as the perimeter or area.Sometimes, if you know the perimeter and two side lengths, you can find the third side by rearranging the perimeter formula:\[a + b + c = P\]Therefore, if you have the perimeter and two sides, you'd solve for the third side like this:\[c = P - a - b\]In different scenarios, such as using the triangle inequality: the sum of any two sides must be greater than the third side, which helps ensure that the given lengths can form a triangle. Knowing how to manipulate these formulas is handy for solving more complex mathematical and real-world problems related to triangles.