Problem 56 - Prealgebra

Question

GEOMETRY The perimeter of a rectangle is 22 feet. Its length is 7 feet. Find its width. (Lesson \(3-8\) )

Step-by-Step Solution

Answer

The width of the rectangle is 4 feet.

Step 1: Understand the problem

We need to find the width of a rectangle. The problem gives us the perimeter, which is 22 feet, and the length, which is 7 feet.

Step 2: Recall perimeter formula for a rectangle

The formula for the perimeter of a rectangle is given by: \[ P = 2 imes ( ext{length} + ext{width}) \]in this case, \( P = 22 \text{ feet} \) and length \( = 7 \text{ feet} \).

Step 3: Set up the equation with known values

Substitute the known values into the perimeter formula: \[ 22 = 2 imes (7 + \text{width}) \].

Step 4: Simplify the equation

First, divide both sides of the equation by 2 to simplify: \[ 11 = 7 + \text{width} \].

Step 5: Solve for the width

Subtract 7 from both sides of the equation to isolate the width: \[ \text{width} = 11 - 7 \].

Step 6: Calculate the width

Perform the subtraction to find the width: \[ \text{width} = 4 \text{ feet} \].

Key Concepts

Understanding Geometry Problems

In geometry, a problem often requires finding unknown measurements. Knowing how to approach and break down these problems is crucial. It starts with identifying what we know and what we need to find. For this rectangle problem:

• We know the perimeter is 22 feet.
• We are provided with the length, which is 7 feet.
• We need to find the width.

Recognizing these details helps us apply the right formulas and methods. Geometry problems like this one help us understand relationships between different properties of shapes. Breaking down problems into smaller parts, like translating words into mathematical equations, is often key to solving them successfully. Always refer back to what is being asked, ensuring every known value is correctly used.

Width Calculation in a Rectangle

Calculating the width of a rectangle involves understanding the formula for the perimeter, which is the total distance around the rectangle. For rectangles, this is formulated as: \[ P = 2 \times ( \text{length} + \text{width} ) \]Given a perimeter, solving for width requires rearranging this formula. Here’s how we handle the calculation step-by-step:

• Insert the given perimeter and length into the formula: \[ 22 = 2 \times (7 + \text{width}) \]
• Divide both sides by 2 to simplify: \[ 11 = 7 + \text{width} \]
• Finally, solve for the width by isolating it: \[ \text{width} = 11 - 7 \]
• Calculate: the width equals 4 feet.

This step-by-step method is reliable for finding missing dimensions when certain values of a rectangle are already known.

Rectangle Properties

Rectangles are four-sided polygons, classified as quadrilaterals, with particular properties that define them. Understanding these properties helps in multiple geometry problems. Here are the key characteristics of rectangles:

• Opposite sides are equal in length and parallel.
• All angles are right angles, meaning they each measure 90 degrees.
• The diagonals are equal in length.

These properties ensure that when you know the length and the perimeter, you can always find the width, as demonstrated in the exercise problem. The relationships between perimeter, length, and width highlight how these properties support basic geometry problem-solving. Knowing these can ease the process of tackling more complex concepts involving rectangles and other geometric shapes.