9.5: Use Properties of Rectangles, Triangles, and Trapezoids
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By the end of this section, you will be able to:
- Understand linear, square, and cubic measure
- Use properties of rectangles
- Use properties of triangles
- Use properties of trapezoids
Be Prepared 9.10
Before you get started, take this readiness quiz.
The length of a rectangle is less than the width. Let represent the width. Write an expression for the length of the rectangle.
If you missed this problem, review Example 2.26.
Be Prepared 9.11
Simplify:
If you missed this problem, review Example 7.7.
Be Prepared 9.12
Simplify:
If you missed this problem, review Example 5.36.
In this section, we’ll continue working with geometry applications. We will add some more properties of triangles, and we’ll learn about the properties of rectangles and trapezoids.
Understand Linear, Square, and Cubic Measure
When you measure your height or the length of a garden hose, you use a ruler or tape measure (Figure 9.13). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimeter and meter are units of linear measure.
When you want to know how much tile is needed to cover a floor, or the size of a wall to be painted, you need to know the area, a measure of the region needed to cover a surface. Area is measured is square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm) on each side. A square inch is a square that is one inch on each side (Figure 9.14).
Figure 9.15 shows a rectangular rug that is feet long by feet wide. Each square is foot wide by foot long, or square foot. The rug is made of squares. The area of the rug is square feet.
When you measure how much it takes to fill a container, such as the amount of gasoline that can fit in a tank, or the amount of medicine in a syringe, you are measuring volume. Volume is measured in cubic units such as cubic inches or cubic centimeters. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimeters, cubic inches, and cubic feet. A cubic centimeter is a cube that measures one centimeter on each side, while a cubic inch is a cube that measures one inch on each side (Figure 9.16).
Suppose the cube in Figure 9.17 measures inches on each side and is cut on the lines shown. How many little cubes does it contain? If we were to take the big cube apart, we would find little cubes, with each one measuring one inch on all sides. So each little cube has a volume of cubic inch, and the volume of the big cube is cubic inches.
Manipulative Mathematics
Example 9.25
For each item, state whether you would use linear, square, or cubic measure:
- ⓐ amount of carpeting needed in a room
- ⓑ extension cord length
- ⓒ amount of sand in a sandbox
- ⓓ length of a curtain rod
- ⓔ amount of flour in a canister
- ⓕ size of the roof of a doghouse.
- Answer
ⓐ You are measuring how much surface the carpet covers, which is the area. square measure ⓑ You are measuring how long the extension cord is, which is the length. linear measure ⓒ You are measuring the volume of the sand. cubic measure ⓓ You are measuring the length of the curtain rod. linear measure ⓔ You are measuring the volume of the flour. cubic measure ⓕ You are measuring the area of the roof. square measure
Try It 9.49
Determine whether you would use linear, square, or cubic measure for each item.
ⓐ amount of paint in a can ⓑ height of a tree ⓒ floor of your bedroom ⓓ diameter of bike wheel ⓔ size of a piece of sod ⓕ amount of water in a swimming pool
Try It 9.50
Determine whether you would use linear, square, or cubic measure for each item.
ⓐ volume of a packing box ⓑ size of patio ⓒ amount of medicine in a syringe ⓓ length of a piece of yarn ⓔ size of housing lot ⓕ height of a flagpole
Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean.
Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor.
Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter.
Perimeter and Area
The perimeter is a measure of the distance around a figure.
The area is a measure of the surface covered by a figure.
Figure 9.18 shows a square tile that is inch on each side. If an ant walked around the edge of the tile, it would walk inches. This distance is the perimeter of the tile.
Since the tile is a square that is inch on each side, its area is one square inch. The area of a shape is measured by determining how many square units cover the shape.
Manipulative Mathematics
Example 9.26
Each of two square tiles is square inch. Two tiles are shown together.
- ⓐ What is the perimeter of the figure?
- ⓑ What is the area?
- Answer
ⓐ The perimeter is the distance around the figure. The perimeter is inches.
ⓑ The area is the surface covered by the figure. There are square inch tiles so the area is square inches.
Try It 9.51
Each box in the figure below is 1 square inch. Find the ⓐ perimeter and ⓑ area of the figure:
Try It 9.52
Each box in the figure below is 1 square inch. Find the ⓐ perimeter and ⓑ area of the figure:
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,
The perimeter, of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk units, or two lengths and two widths. The perimeter then is
What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was
Properties of Rectangles
- Rectangles have four sides and four right
angles.( 90° ) ( 90° ) - The lengths of opposite sides are equal.
- The perimeter,
Figure 9.19.P , P = 2 L + 2 W P = 2 L + 2 W - The area,
of a rectangle is the length times the width.A , A , A = L ⋅ W 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.
How To
Use a Problem Solving Strategy for Geometry Applications
- Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
- Step 2. Identify what you are looking for.
- Step 3. Name what you are looking for. Choose a variable to represent that quantity.
- Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Step 5. Solve the equation using good algebra techniques.
- Step 6. Check the answer in the problem and make sure it makes sense.
- Step 7. Answer the question with a complete sentence.
Example 9.27
The length of a rectangle is
- Answer
ⓐ 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 640 square meters.
Try It 9.53
The length of a rectangle is
Try It 9.54
The length of a rectangle is
Example 9.28
Find the length of a rectangle with perimeter
- Answer
Step 1. Read the problem. Draw the figure and label it with the given information. 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.
Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. The length is 15 inches.
Try It 9.55
Find the length of a rectangle with a perimeter of
Try It 9.56
Find the length of a rectangle with a perimeter of
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 9.29
The width of a rectangle is two inches less than the length. The perimeter is
- Answer
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
Step 4.Translate.
Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information.
Substitute in the given information.
Step 5. Solve the equation. 52 = 2 L + 2 L − 4 52 = 2 L + 2 L − 4 Combine like terms. 52 = 4 L − 4 52 = 4 L − 4 Add 4 to each side. 56 = 4 L 56 = 4 L Divide by 4. 56 4 = 4 L 4 56 4 = 4 L 4 14 = L 14 = L The length is 14 inches. Now we need to find the width. The width is L − 2.
The width is 12 inches.Step 6. Check:
Since , this works!14 + 12 + 14 + 12 = 52 14 + 12 + 14 + 12 = 52 Step 7. Answer the question. The length is 14 feet and the width is 12 feet.
Try It 9.57
The width of a rectangle is seven meters less than the length. The perimeter is
Try It 9.58
The length of a rectangle is eight feet more than the width. The perimeter is
Example 9.30
The length of a rectangle is four centimeters more than twice the width. The perimeter is
- Answer
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
Step 4.Translate.
Write the appropriate formula and substitute in the given information.Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. The length is 12 cm and the width is 4 cm.
Try It 9.59
The length of a rectangle is eight more than twice the width. The perimeter is
Try It 9.60
The width of a rectangle is six less than twice the length. The perimeter is
Example 9.31
The area of a rectangular room is
- Answer
Step 1. Read the problem. 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.Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. The width of the room is 12 feet.
Try It 9.61
The area of a rectangle is
Try It 9.62
The width of a rectangle is
Example 9.32
The perimeter of a rectangular swimming pool is
- Answer
Step 1. Read the problem. Draw the figure and label it with the given information. 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 = width W = width
W + 15 = length W + 15 = length Step 4.Translate.
Write the appropriate formula and substitute.Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. The length of the pool is 45 feet and the width is 30 feet.
Try It 9.63
The perimeter of a rectangular swimming pool is
Try It 9.64
The length of a rectangular garden is
Use the Properties of Triangles
We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in Figure 9.20, we’ve labeled the length
We can divide this rectangle into two congruent triangles (Figure 9.22). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or
The formula for the area of a triangle is
To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a
Triangle Properties
For any triangle
The perimeter of a triangle is the sum of the lengths of the sides.
The area of a triangle is one-half the base,
Example 9.33
Find the area of a triangle whose base is
- Answer
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 the triangle Step 3. Name. Choose a variable to represent it. let A = area of the triangle Step 4.Translate.
Write the appropriate formula.
Substitute.
Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. The area is 44 square inches.
Try It 9.65
Find the area of a triangle with base
Try It 9.66
Find the area of a triangle with base
Example 9.34
The perimeter of a triangular garden is
- Answer
Step 1. Read the problem. Draw the figure and label it with the given information. Step 2. Identify what you are looking for. length of the third side of a triangle Step 3. Name. Choose a variable to represent it. Let c = the third side Step 4.Translate.
Write the appropriate formula.
Substitute in the given information.
Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. The third side is 11 feet long.
Try It 9.67
The perimeter of a triangular garden is
Try It 9.68
The lengths of two sides of a triangular window are
Example 9.35
The area of a triangular church window is
- Answer
Step 1. Read the problem. Draw the figure and label it with the given information. Step 2. Identify what you are looking for. height of a triangle Step 3. Name. Choose a variable to represent it. Let h = the height Step 4.Translate.
Write the appropriate formula.
Substitute in the given information.
Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. The height of the triangle is 12 meters.
Try It 9.69
The area of a triangular painting is
Try It 9.70
A triangular tent door has an area of
Isosceles and Equilateral Triangles
Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. Figure 9.24 shows both types of triangles.
Isosceles and Equilateral Triangles
An isosceles triangle has two sides the same length.
An equilateral triangle has three sides of equal length.
Example 9.36
The perimeter of an equilateral triangle is
- Answer
Step 1. Read the problem. Draw the figure and label it with the given information.
Perimeter = 93 in.Step 2. Identify what you are looking for. length of the sides of an equilateral triangle Step 3. Name. Choose a variable to represent it. Let s = length of each side Step 4.Translate.
Write the appropriate formula.
Substitute.
Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. Each side is 31 inches.
Try It 9.71
Find the length of each side of an equilateral triangle with perimeter
Try It 9.72
Find the length of each side of an equilateral triangle with perimeter
Example 9.37
Arianna has
- Answer
Step 1. Read the problem. Draw the figure and label it with the given information.
P = 156 in.Step 2. Identify what you are looking for. the lengths of the two equal sides Step 3. Name. Choose a variable to represent it. Let s = the length of each side Step 4.Translate.
Write the appropriate formula.
Substitute in the given information.
Step 5. Solve the equation. Step 6. Check:
Step 7. Answer the question. Arianna can make each of the two equal sides 48 inches long.
Try It 9.73
A backyard deck is in the shape of an isosceles triangle with a base of
Try It 9.74
A boat’s sail is an isosceles triangle with base of
Use the Properties of Trapezoids
A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base
The formula for the area of a trapezoid is:
Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See Figure 9.26.
The height of the trapezoid is also the height of each of the two triangles. See Figure 9.27.
The formula for the area of a trapezoid is
If we distribute, we get,
Properties of Trapezoids
- A trapezoid has four sides. See Figure 9.25.
- Two of its sides are parallel and two sides are not.
- The area,
of a trapezoid isA , A , .A = 1 2 h ( b + B ) A = 1 2 h ( b + B )
Example 9.38
Find the area of a trapezoid whose height is 6 inches and whose bases are
- Answer
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 the trapezoid Step 3. Name. Choose a variable to represent it. Let A = the area A = the area Step 4.Translate.
Write the appropriate formula.
Substitute.
Step 5. Solve the equation. Step 6. Check: Is this answer reasonable? If we draw a rectangle around the trapezoid that has the same big base
and a heightB B its area should be greater than that of the trapezoid.h , h , If we draw a rectangle inside the trapezoid that has the same little base
and a heightb b its area should be smaller than that of the trapezoid.h , h , The area of the larger rectangle is
square inches and the area of the smaller rectangle is84 84 square inches. So it makes sense that the area of the trapezoid is between66 66 and84 84 square inches66 66 Step 7. Answer the question. The area of the trapezoid is
square inches.75 75
Try It 9.75
The height of a trapezoid is
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The height of a trapezoid is
Example 9.39
Find the area of a trapezoid whose height is
- Answer
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 the trapezoid 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: Is this answer reasonable?
The area of the trapezoid should be less than the area of a rectangle with base 13.7 and height 5, but more than the area of a rectangle with base 10.3 and height 5.
Step 7. Answer the question. The area of the trapezoid is 60 square feet.
Try It 9.77
The height of a trapezoid is
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The height of a trapezoid is
Example 9.40
Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of
- Answer
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 trapezoid 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: Is this answer reasonable?
Yes. The area of the trapezoid is less than the area of a rectangle with a base of 8.2 yd and height 3.4 yd, but more than the area of a rectangle with base 5.6 yd and height 3.4 yd.
Step 7. Answer the question. Vinny has 23.46 square yards in which he can plant.
Try It 9.79
Lin wants to sod his lawn, which is shaped like a trapezoid. The bases are
Try It 9.80
Kira wants cover his patio with concrete pavers. If the patio is shaped like a trapezoid whose bases are
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Section 9.4 Exercises
Practice Makes Perfect
Understand Linear, Square, and Cubic Measure
In the following exercises, determine whether you would measure each item using linear, square, or cubic units.
amount of water in a fish tank
length of dental floss
living area of an apartment
floor space of a bathroom tile
height of a doorway
capacity of a truck trailer
In the following exercises, find the ⓐ perimeter and ⓑ area of each figure. Assume each side of the square is
Use the Properties of Rectangles
In the following exercises, find the ⓐ perimeter and ⓑ area of each rectangle.
The length of a rectangle is
The length of a rectangle is
A rectangular room is
A driveway is in the shape of a rectangle
In the following exercises, solve.
Find the length of a rectangle with perimeter
Find the length of a rectangle with perimeter
Find the width of a rectangle with perimeter
Find the width of a rectangle with perimeter
The area of a rectangle is
The area of a rectangle is
The length of a rectangle is
The width of a rectangle is
The perimeter of a rectangle is
The perimeter of a rectangle is
The width of the rectangle is
The length of the rectangle is
The perimeter of a rectangle of
The length of a rectangle is three times the width. The perimeter is
The length of a rectangle is
The length of a rectangle is
The width of a rectangular window is
The length of a rectangular poster is
The area of a rectangular roof is
The area of a rectangular tarp is
The perimeter of a rectangular courtyard is
The perimeter of a rectangular painting is
The width of a rectangular window is
The width of a rectangular playground is
Use the Properties of Triangles
In the following exercises, solve using the properties of triangles.
Find the area of a triangle with base
Find the area of a triangle with base
Find the area of a triangle with base
Find the area of a triangle with base
A triangular flag has base of
A triangular window has base of
If a triangle has sides of
If a triangle has sides of
What is the base of a triangle with an area of
What is the height of a triangle with an area of
The perimeter of a triangular reflecting pool is
A triangular courtyard has perimeter of
An isosceles triangle has a base of
An isosceles triangle has a base of
Find the length of each side of an equilateral triangle with a perimeter of
Find the length of each side of an equilateral triangle with a perimeter of
The perimeter of an equilateral triangle is
The perimeter of an equilateral triangle is
The perimeter of an isosceles triangle is
The perimeter of an isosceles triangle is
A dish is in the shape of an equilateral triangle. Each side is
A floor tile is in the shape of an equilateral triangle. Each side is
A road sign in the shape of an isosceles triangle has a base of
A scarf in the shape of an isosceles triangle has a base of
The perimeter of a triangle is
The perimeter of a triangle is
One side of a triangle is twice the smallest side. The third side is
One side of a triangle is three times the smallest side. The third side is
Use the Properties of Trapezoids
In the following exercises, solve using the properties of trapezoids.
The height of a trapezoid is
The height of a trapezoid is
Find the area of a trapezoid with a height of
Find the area of a trapezoid with a height of
The height of a trapezoid is
The height of a trapezoid is
Find the area of a trapezoid with a height of
Find the area of a trapezoid with a height of
Laurel is making a banner shaped like a trapezoid. The height of the banner is
Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with width
Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width
Elena is knitting a scarf. The scarf will be shaped like a trapezoid with width
Everyday Math
Fence Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a
Gardening Lupita wants to fence in her tomato garden. The garden is rectangular and the length is twice the width. It will take
Fence Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are
Painting Caleb wants to paint one wall of his attic. The wall is shaped like a trapezoid with height
Writing Exercises
If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.
If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning.
Look at the two figures.
ⓐ Which figure looks like it has the larger area? Which looks like it has the larger perimeter?
ⓑ Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter?
The length of a rectangle is
ⓐ Write the equation you would use to solve the problem.
ⓑ Why can’t you solve this equation with the methods you learned in the previous chapter?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?