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9.4 Use properties of rectangles, triangles, and trapezoids  (Page 3/17)

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Find the ⓐ perimeter and ⓑ area of the figure:

A rectangle is shown comprised of 3 squares.

  1. ⓐ 8 inches
  2. ⓑ 3 sq. inches

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Find the ⓐ perimeter and ⓑ area of the figure:

A square is shown comprised of 4 smaller squares.

  1. ⓐ 8 centimeters
  2. ⓑ 4 sq. centimeters

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Use 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 [link] .

A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled L, the sides are labeled W.
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

P = L + W + L + W or P = 2 L + 2 W

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 [link] . Since A = 2 ⋅ 3 , we see that the area, A , is the length, L , times the width, W , so the area of a rectangle is A = L ⋅ W .

A rectangle is shown. It is made up of 6 squares. The bottom is 2 squares across and marked as 2, the side is 3 squares long and marked as 3.
The area of this rectangular rug is 6 square feet, its length times its width.

Properties of rectangles

  • Rectangles have four sides and four right ( 90° ) 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 [link] .
    P = 2 L + 2 W
  • The area, A , of a rectangle is the length times the width.
    A = L ⋅ W

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

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.

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. .
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.
.
Step 5. Solve the equation. .
Step 6. Check:
.
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. .
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.
.
Step 5. Solve the equation. .
Step 6. Check:
.
Step 7. Answer the question. The area of the rectangle is 60 square meters.
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The length of a rectangle is 120 yards and the width is 50 yards. Find ⓐ the perimeter and ⓑ the area.

  1. ⓐ 340 yd
  2. ⓑ 6000 sq. yd

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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