Problem 12 - Intermediate Algebra

Question

Find the perimeter of each figure. See Example 1. A triangle with sides \(1.8,1.8,\) and \(1.5 \mathrm{cm}\) long

Step-by-Step Solution

Answer

The perimeter of the triangle is 5.1 cm.

Step 1: Understanding the Formula

The perimeter of a geometric figure is the sum of the lengths of all its sides. For a triangle with sides of lengths \(a\), \(b\), and \(c\), the formula to find the perimeter is: \[ P = a + b + c \]

Step 2: Identifying the Side Lengths

We have a triangle with sides of lengths \(1.8\) cm, \(1.8\) cm, and \(1.5\) cm.

Step 3: Apply the Formula

Substitute the side lengths into the perimeter formula:\[ P = 1.8 + 1.8 + 1.5 \]

Step 4: Calculation

Perform the addition: \[ 1.8 + 1.8 = 3.6 \] Then add \(1.5\): \[ 3.6 + 1.5 = 5.1 \]

Step 5: Conclusion

The perimeter of the triangle is \(5.1\) cm. Ensure the unit is correctly noted as centimeters (cm).

Key Concepts

Geometry

Geometry is a fascinating branch of mathematics that explores shapes, sizes, and the properties of space. Among other structures, it deals with figures like triangles, circles, and polygons. Understanding geometry helps us navigate both the physical and abstract world around us. In simple terms, while geometry might sound complex, it's all about the figures and forms you encounter every day.

• It helps you understand how objects fit together.
• Geometry is everywhere—from architecture to art.
• It uses points, lines, surfaces, and solids to form different shapes.

Learning geometry enhances spatial understanding, which is crucial for numerous daily activities, artistic pursuits, and even professional tasks like engineering and design.

Triangle

Triangles are one of the simplest yet intriguing shapes in geometry. They have three sides, three vertices, and three angles. All the internal angles of a triangle add up to 180 degrees. This makes triangles a fundamental shape in various geometrical constructions.
A triangle can be identified by its side lengths and angles. Here's a quick overview of triangle types based on their sides:

• **Equilateral Triangle**: All sides are equal in length.
• **Isosceles Triangle**: Two sides are of equal length.
• **Scalene Triangle**: All sides have different lengths.

Triangles are not just theoretical; they are prominent in real-world structures like bridges and towers because they provide stability and strength. Recognizing the type of triangle you are working with can simplify calculations like finding the perimeter.

Perimeter Formula

The perimeter of a shape, such as a triangle, is the total distance around its edges. For triangles, you calculate the perimeter by adding up the lengths of all three sides. This concept is straightforward but critical for many practical applications. The formula for calculating the perimeter is:\[ P = a + b + c \]Where \(a\), \(b\), and \(c\) are the lengths of the triangle's sides.
Using this formula:

• Ensure all side lengths are measured in the same units.
• Adding these values together will give you the perimeter.
• This sum tells you the total boundary length of the triangle.

Calculating the perimeter is useful not only in mathematics but also in real-life situations—for instance, determining how much material is needed to encircle a plot of land. Remember, the correct and consistent use of units in calculations ensures accuracy, a vital skill both in class and in practical scenarios.