Problem 39 - Prealgebra

Question

Draw and label a rectangle that has a perimeter of 18 inches.

Step-by-Step Solution

Answer

Draw a rectangle with any length and width pair like 5 and 4 inches where the sum of length and width is 9, ensuring the perimeter is 18 inches.

Step 1: Understand the Problem

A rectangle has two pairs of equal sides. The perimeter is the total distance around the rectangle. For a rectangle with a length of \( l \) inches and a width of \( w \) inches, the perimeter \( P \) is given by the formula: \( P = 2(l + w) \). We need to find length and width such that \( P = 18 \) inches.

Step 2: Set Up the Equation

Using the formula for perimeter, set up the equation: \( 2(l + w) = 18 \). Simplify this equation to find a relationship between \( l \) and \( w \): \[ l + w = 9 \]

Step 3: Choose Dimensions

Choose any positive integer values for \( l \) and \( w \) that satisfy \( l + w = 9 \). Examples could be \( l = 4 \) and \( w = 5 \) or \( l = 6 \) and \( w = 3 \).

Step 4: Draw the Rectangle

On a piece of paper, draw a rectangle where the longer side represents the length \( l \) and the shorter side represents the width \( w \). Clearly label the sides of your rectangle with the selected values. For example, if you chose \( l = 5 \) and \( w = 4 \), label one pair of opposite sides as 5 inches and the other pair as 4 inches.

Step 5: Verify the Perimeter

Add the lengths of all sides to ensure the perimeter is 18 inches. For example, using \( l = 5 \) and \( w = 4 \): the perimeter is \( 2(5 + 4) = 18 \) inches. Confirm that your chosen dimensions are correct.

Key Concepts

Rectangle Dimensions

Understanding rectangle dimensions is crucial in geometry. A rectangle is characterized by having four sides, with opposite sides being equal in length. These sides are typically referred to as the length (longer side) and the width (shorter side). In any rectangle, the dimensions are expressed by these two measurements:

• Length (l): This is the longer side of the rectangle.
• Width (w): This is the shorter side of the rectangle.

When discussing rectangle dimensions, learners should remember that changing the values of length or width will affect not only the shape but also the perimeter and area of the rectangle. Therefore, identifying and labeling these dimensions accurately is fundamental when solving mathematical problems involving rectangles.

Perimeter Formula

In geometry, the perimeter of a rectangle is the total length around the shape. This is calculated using the perimeter formula, which involves both the length and the width:

The formula for the perimeter (\(P\)) of a rectangle is:

• \( P = 2(l + w) \)

This formula accounts for all sides of the rectangle, meaning both pairs of opposite sides are considered: one pair of lengths and one pair of widths. The perimeter formula is essential for tasks such as fencing a yard or crafting anything rectangular. Thus, practicing its application in various exercises helps deepen students' understanding of geometry.

Mathematical Problem-Solving

Efficient problem-solving in mathematics requires a step-by-step approach. In the case of finding a rectangle's dimensions given a perimeter, students should:

• Understand the problem by identifying what the perimeter means and how it relates to length and width.
• Set up the correct equation using the perimeter formula.
• Use logical reasoning to choose possible values for length and width that satisfy the equation.
• Verify the solution by checking calculations and confirming they make sense within the context of the problem.

These steps are fundamental in developing mathematical problem-solving skills. They allow students to approach problems systematically, ensuring they arrive at correct and logical conclusions.

Geometry Concepts

Geometry offers insight into various shapes and their properties, such as angles, dimensions, and boundaries. In the case of a rectangle, several geometric concepts are interlinked:

• Parallel Sides: A rectangle's opposite sides are parallel and equal in length.
• Right Angles: Each corner of a rectangle forms a right angle (90 degrees).
• Symmetry: Rectangles possess lines of symmetry that reflect equal halves.

Understanding these geometry concepts helps in visualizing and solving problems related to rectangles. It adds to the learner's broader comprehension of geometric principles, which is vital for higher-level mathematics and practical real-world applications.