Problem 14 - Algebra 1

Question

perimeter \(=3 \mathrm{cm}+4 \mathrm{cm}+5 \mathrm{cm}\)

Step-by-Step Solution

Answer

The perimeter of the triangle is 12 cm.

Step 1: Identify the given lengths

Here the sides of the triangle are given as 3 cm, 4 cm, and 5 cm.

Step 2: Write down the formula for the perimeter

The formula for calculating the perimeter of a triangle is \(P=a + b + c\), where \(a,b,c\) are the lengths of each side of the triangle.

Step 3: Substituting the given values into the formula

Substitute the lengths of the sides into the formula: \(P=3 cm + 4 cm + 5 cm\).

Step 4: Calculate the perimeter

Now, add the given lengths: \(P = 12 cm\).

Key Concepts

Understanding Triangles

A triangle is a basic geometric shape with three sides and three corners, also known as vertices. Triangles are part of the larger family of polygons and are fundamental shapes in geometry. Each triangle has three sides that can vary in length, leading to different types of triangles:

• Equilateral Triangle: All sides are the same length.
• Isosceles Triangle: Two sides are of equal length.
• Scalene Triangle: All sides are of different lengths.

Recognizing these types is important for solving problems related to the perimeter. In the exercise, the example is a scalene triangle with side lengths of 3 cm, 4 cm, and 5 cm. Triangles also have unique angle properties like additive angles summing to 180 degrees, which can sometimes help in solving perimeter-related tasks by understanding the overall shape better.

Basic Principles of Geometry

Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, and shapes. It forms the foundation for understanding spatial relationships. When we talk about the geometry of a triangle, we refer to aspects like side lengths, angles, and area. One useful geometric concept is that the perimeter is a measure of the distance around the boundary of a shape. This concept helps us understand and calculate the perimeter of any polygon, including triangles. Knowing how to apply geometrical concepts is vital for measurements, design, and various applications in science and engineering. In our example, geometry basics help identify the triangle sides and calculate the perimeter using the sum of these sides. This encourages clear and methodical calculations, which are essential skills in geometry as they ensure correct results, especially in exams and practical applications.

Mathematical Formula for Perimeter

The mathematical formula for finding the perimeter of a triangle is straightforward but essential. The formula is: \[ P = a + b + c \] - Here, \(P\) stands for perimeter, and \(a, b,\) and \(c\) are the lengths of the sides of the triangle. - Calculating the perimeter using this formula involves adding the lengths of all sides, which measures the total boundary around the triangle.In the provided example, substituting the side lengths of 3 cm, 4 cm, and 5 cm into the formula gives: \[ P = 3 \, \text{cm} + 4 \, \text{cm} + 5 \, \text{cm} = 12 \, \text{cm} \] By applying this formula, you can quickly determine the perimeter of any triangle once the side lengths are known. This demonstrates the power and simplicity of mathematical formulas, helping break down complex problems into simple, understandable calculations.