Problem 86 - Algebra for College Students

Question

Answer the question with an algebraic expression. The perimeter of a square is \(c\) centimeters. How long is each side of the square?

Step-by-Step Solution

Answer

The side length is \(\frac{c}{4}\) centimeters.

Step 1: Understanding the Perimeter Formula

The formula for the perimeter of a square is given as the sum of all its sides. Since a square has four equal sides, the perimeter can be expressed as \(4s\), where \(s\) is the length of one side.

Step 2: Set Up the Equation

From the problem, we know the perimeter is \(c\) centimeters. Therefore, we can set up the equation for the perimeter as \(4s = c\).

Step 3: Solve for the Side Length

To find the side length \(s\), we need to solve the equation \(4s = c\) for \(s\). Divide both sides by 4 to isolate \(s\): \(s = \frac{c}{4}\).

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations like addition or multiplication. In the context of the given exercise, we use an algebraic expression to define the perimeter of a square. A square's perimeter can be described as the sum of all its sides. Since each side is equal, the expression becomes 4 times the length of one side, or \( 4s \). Here, \( s \) represents the side length, and \( c \) is the total perimeter. Using algebraic expressions helps encapsulate relationships in mathematical problems, such as defining how the perimeter relates to side length in this scenario.

• Algebraic expressions include variables and numbers.
• They help in expressing relationships succinctly.
• In geometry, they can describe dimensions and measures.

By forming and manipulating these expressions, we can effectively solve problems and make predictions about geometric figures.

Solving Equations

Solving equations involves finding the value of the variables that make the equation true. Here, the task is to find out what one side of the square measures given the perimeter. We start with our equation, \( 4s = c \), from our algebraic expression. The goal in solving this equation is to isolate \( s \), which represents the side length of the square.To extract \( s \), we perform the same operation on both sides of the equation to maintain equality. Dividing both sides by 4, we obtain the equation \( s = \frac{c}{4} \). This step ensures the variable \( s \) is by itself on one side of the equation, giving us the side length in terms of the perimeter.

• Understanding how to manipulate equations is critical in algebra.
• Isolating the variable helps find specific measures and solutions.
• Dividing or multiplying both sides by the same number keeps equations balanced.

Using these basic steps to solve equations enables us to explore and find unknowns in mathematical problems effectively.

Geometry

Geometry is the branch of mathematics concerning shapes, sizes, and the properties of space. In the given problem, we're focused on a square, a simple geometric shape with unique properties. The main feature of a square is that all its sides are equal, and it has four right angles. The concept of perimeter in geometry refers to the total distance around a shape. For a square, this translates to four times the length of one side. The study of geometry allows us to apply algebra, as we did with algebraic expressions, to solve real-world and theoretical problems. For instance, knowing the perimeter, you can find side lengths, see relationships between other geometric figures, and apply it to various fields, from architecture to engineering.

• Geometry helps in understanding and solving spatial problems.
• Relates algebraic expressions to actual dimensions and physical spaces.
• Supports visual learning through figures and illustrations.

By understanding geometric principles, one can navigate and solve problems involving space and dimensions more effectively.