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Using the Properties of Rectangles

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Using the Properties of Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, \(L,\) and the adjacent side as the width, \(W.\) See the figure below.

Using the Properties of Rectangles

A rectangle has four sides, and four right angles. The sides are labeled L for length and W for width.

The perimeter, \(P,\) of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk \(L+W+L+W\) units, or two lengths and two widths. The perimeter then is

\(\begin{array}{c}P=L+W+L+W\hfill \\ \hfill \text{or}\hfill \\ P=2L+2W\hfill \end{array}\)

What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was \(2\) feet long by \(3\) feet wide, and its area was \(6\) square feet. See the figure below. Since \(A=2\cdot 3,\) we see that the area, \(A,\) is the length, \(L,\) times the width, \(W,\) so the area of a rectangle is \(A=L\cdot W.\)

Using the Properties of Rectangles

The area of this rectangular rug is \(6\) square feet, its length times its width.

Definition: Properties of Rectangles

  • Rectangles have four sides and four right \(\left(\text{90°}\right)\) angles.
  • The lengths of opposite sides are equal.
  • The perimeter, \(P,\) of a rectangle is the sum of twice the length and twice the width. See the figure below.
    \(P=2L+2W\)

  • The area, \(A,\) of a rectangle is the length times the width.
    \(A=L\cdot W\)

For easy reference as we work the examples in this section, we will restate the Problem Solving Strategy for Geometry Applications here.

How to Use a Problem Solving Strategy for Geometry Applications

  1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
  2. Identify what you are looking for.
  3. Name what you are looking for. Choose a variable to represent that quantity.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.

Optional Video: Perimeter of a Rectangle

Example

The length of a rectangle is \(32\) meters and the width is \(20\) meters. Find the perimeter, and the area.

Solution

  
Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Rectangles
Step 2. Identify what you are looking for.the perimeter of a rectangle
Step 3. Name. Choose a variable to represent it.Let P = the perimeter
Step 4. Translate.

 

Write the appropriate formula.

 

Substitute.

 

Using the Properties of Rectangles

Step 5. Solve the equation.Using the Properties of Rectangles
Step 6. Check:

 

Using the Properties of Rectangles

 
Step 7. Answer the question.The perimeter of the rectangle is 104 meters.
  
Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Rectangles
Step 2. Identify what you are looking for.the area of a rectangle
Step 3. Name. Choose a variable to represent it.Let A = the area
Step 4. Translate.

 

Write the appropriate formula.

 

Substitute.

 

Using the Properties of Rectangles

Step 5. Solve the equation.Using the Properties of Rectangles
Step 6. Check:

 

Using the Properties of Rectangles

 
Step 7. Answer the question.The area of the rectangle is 60 square meters.

Example

Find the length of a rectangle with perimeter \(50\) inches and width \(10\) inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Rectangles
Step 2. Identify what you are looking for.the length of the rectangle
Step 3. Name. Choose a variable to represent it.Let L = the length
Step 4. Translate.

 

Write the appropriate formula.

 

Substitute.

 

Using the Properties of Rectangles

Step 5. Solve the equation.Using the Properties of Rectangles
Step 6. Check:

 

Using the Properties of Rectangles

 
Step 7. Answer the question.The length is 15 inches.

In the next example, the width is defined in terms of the length. We’ll wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

Example

The width of a rectangle is two inches less than the length. The perimeter is \(52\) inches. Find the length and width.

Solution

Step 1. Read the problem. 
Step 2. Identify what you are looking for.the length and width of the rectangle
Step 3. Name. Choose a variable to represent it.

 
 

Now we can draw a figure using these expressions for the length and width.

Since the width is defined in terms of the length, we let L = length. The width is two feet less that the length, so we let L − 2 = width

 

Using the Properties of Rectangles

Step 4.Translate.

 

Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information.

 

Substitute in the given information.

 

Using the Properties of Rectangles

Step 5. Solve the equation.\(52=2L+2L-4\)
Combine like terms.\(52=4L-4\)
Add 4 to each side.\(56=4L\)
Divide by 4.\(\frac{56}{4}=\frac{4L}{4}\)
 \(14=L\)
 The length is 14 inches.
Now we need to find the width. 
The width is L − 2.Using the Properties of Rectangles

 

The width is 12 inches.

Step 6. Check:

 

Since \(14+12+14+12=52\), this works!

 
Step 7. Answer the question.The length is 14 feet and the width is 12 feet.

Example

The length of a rectangle is four centimeters more than twice the width. The perimeter is \(32\) centimeters. Find the length and width.

Solution

Step 1. Read the problem. 
Step 2. Identify what you are looking for.the length and width
Step 3. Name. Choose a variable to represent it.let W = width

 

The length is four more than twice the width.

 

2w + 4 = length

 

Using the Properties of Rectangles

Step 4.Translate.

 

Write the appropriate formula and substitute in the given information.

Using the Properties of Rectangles
Step 5. Solve the equation.Using the Properties of Rectangles
Step 6. Check:

 

Using the Properties of Rectangles

 
Step 7. Answer the question.The length is 12 cm and the width is 4 cm.

Example

The area of a rectangular room is \(168\) square feet. The length is \(14\) feet. What is the width?

Solution

Step 1. Read the problem.Using the Properties of Rectangles
Step 2. Identify what you are looking for.the width of a rectangular room
Step 3. Name. Choose a variable to represent it.Let W = width
Step 4.Translate.

 

Write the appropriate formula and substitute in the given information.

Using the Properties of Rectangles
Step 5. Solve the equation.Using the Properties of Rectangles
Step 6. Check:

 

Using the Properties of Rectangles

 
Step 7. Answer the question.The width of the room is 12 feet.

Example

The perimeter of a rectangular swimming pool is \(150\) feet. The length is \(15\) feet more than the width. Find the length and width.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.Using the Properties of Rectangles
Step 2. Identify what you are looking for.the length and width of the pool
Step 3. Name. Choose a variable to represent it.

 

The length is 15 feet more than the width.

Let \(W=\text{width}\)

 

\(W+15=\text{length}\)

Step 4.Translate.

 

Write the appropriate formula and substitute.

Using the Properties of Rectangles
Step 5. Solve the equation.Using the Properties of Rectangles
Step 6. Check:

 

Using the Properties of Rectangles

 
Step 7. Answer the question.The length of the pool is 45 feet and the width is 30 feet.

Optional Video: Area of a Rectangle

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