Problem 43 - Algebra 2

Question

For Exercises 42 and \(43,\) define a variable, write an equation, and solve the problem. GEOMETRY The perimeter of a regular octagon is 124 inches. Find the length of each side.

Step-by-Step Solution

Answer

Each side of the octagon is 15.5 inches long.

Step 1: Define the Variable

Let the length of each side of the regular octagon be denoted by \( x \). Since a regular octagon has 8 sides, the total perimeter can be represented in terms of \( x \).

Step 2: Write the Equation for Perimeter

The perimeter \( P \) of a regular octagon is given by the sum of the lengths of all its sides. Therefore, we can write the equation: \[ P = 8x = 124 \] where 124 inches is the total given perimeter of the octagon.

Step 3: Solve the Equation for x

To find the length of one side \( x \), divide both sides of the equation \( 8x = 124 \) by 8: \[ x = \frac{124}{8} \] On calculating, \( x = 15.5 \). Therefore, the length of each side of the octagon is 15.5 inches.

Key Concepts

Understanding an Octagon

An octagon is a geometrical shape that consists of eight straight sides and eight angles. It is a type of polygon, a collection of points connected by straight lines to form a closed figure. When we refer to a "regular octagon," it means that all sides and angles are equal.
Regular octagons are common, for example, in stop signs or architectural designs, where uniformity is key to aesthetics and balance.
Understanding the properties of an octagon is crucial when solving geometry problems related to calculations of perimeter and area.

Equation Writing

Equation writing is at the core of solving many mathematical problems, including geometry. In the case of determining the perimeter of an octagon, setting up the correct equation is essential.
Here is a simple way to think about it:

• An equation is like a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.
• For the octagon, you're asked to find the length of each side. You represent this unknown quantity with a variable, typically denoted as \(x\).
• Since there are 8 equal sides contributing to the overall perimeter, your equation becomes \(8x = 124\), where 124 is the total perimeter.

This transforms physical properties into a mathematical expression that can then be solved using algebraic methods.

Geometry Problem-Solving

Solving geometry problems often involves a combination of understanding shapes and applying basic math principles. In our octagon example, you need both comprehension of the shape's properties and algebra skills. Here's how you solve such problems smoothly:

• Identify: Recognize what you're asked to find. In this case, the length of each side of the octagon.
• Formulate: Write down what you know in the form of an equation. This involves translating the words "perimeter of a regular octagon is 124 inches" into a math equation \(8x = 124\).
• Solve: Use algebraic techniques to solve the equation. Here, divide 124 by 8 to isolate \(x\), giving you \(x = 15.5\).

Approach each problem methodically, and with practice, the solutions will become intuitive.