3.3: Use Properties of Rectangles, Triangles, and Trapezoids
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- Understand linear, square, and cubic measure
- Use properties of rectangles
- Use properties of triangles
- Use properties of trapezoids
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 3.3.1). 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 3.3.2).
Figure 3.3.3 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 3.3.4).
Suppose the cube in Figure 3.3.5 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.
Example 3.3.1
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
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ⓐ 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
Your Turn 3.3.1
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
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 3.3.6 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.
Example 3.3.2
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
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ⓐ 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.
Your Turn 3.3.2
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, L, and the other side as the width, W.
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 in Figure 3.3.3. It was 6 square feet. Since we see that the area equals the length times the width.
Properties of Rectangles
- Rectangles have four sides and four right angles.
- The lengths of opposite sides are equal.
- The perimeter, , of a rectangle is found by adding all of the sides.
- The area, of a rectangle is the length times the width.
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
- Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
- Identify what you are looking for.
- Name what you are looking for. Choose a variable to represent that quantity.
- Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
Example 3.3.3
The length of a rectangle is meters and the width is meters. Find ⓐ the perimeter, and ⓑ the area.
- Answer
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ⓐ 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.
Your Turn 3.3.3
The length of a rectangle is yards and the width is yards. Find ⓐ the perimeter and ⓑ the area.
Example 3.3.4
Find the length of a rectangle with perimeter inches and width inches.
- 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.
Your Turn 3.3.4
Find the length of a rectangle with a perimeter of inches and width of 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 3.3.5
The width of a rectangle is two inches less than the length. The perimeter is inches. Find the length and width.
- 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. Combine like terms. Add 4 to each side. Divide by 4. 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! Step 7. Answer the question. The length is 14 feet and the width is 12 feet.
Your Turn 3.3.5
The width of a rectangle is seven meters less than the length. The perimeter is meters. Find the length and width.
Example 3.3.6
The length of a rectangle is four centimeters more than twice the width. The perimeter is centimeters. Find the length and width.
- 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.
Your Turn 3.3.6
The length of a rectangle is eight more than twice the width. The perimeter is feet. Find the length and width.
Example 3.3.7
The area of a rectangular room is square feet. The length is feet. What is the width?
- 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.
Your Turn 3.3.7
The area of a rectangle is square feet. The length is feet. What is the width?
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 3.3.8, we’ve labeled the length and the width so it’s area is
We can divide this rectangle into two congruent triangles (Figure 3.3.9). 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 This example helps us see why the formula for the area of a triangle is
The formula for the area of a triangle is where is the base and is the height.
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 angle. Figure 3.3.10 shows three triangles with the base and height of each marked.
Triangle Properties
For any triangle the sum of the measures of the angles is
The perimeter of a triangle is the sum of the lengths of the sides.
The area of a triangle is one-half the base, times the height,
Example 3.3.8
Find the area of a triangle whose base is inches and whose height is inches.
- 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.
Your Turn 3.3.8
Find the area of a triangle with base inches and height inches.
Example 3.3.9
The perimeter of a triangular garden is feet. The lengths of two sides are feet and feet. How long is the third side?
- 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.
Your Turn 3.3.9
The perimeter of a triangular garden is feet. The lengths of two sides are feet and feet. How long is the third side?
Example 3.3.10
The area of a triangular church window is square meters. The base of the window is meters. What is the window’s height?
- 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.
Your Turn 3.3.10
The area of a triangular painting is square inches. The base is inches. What is the height?
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 3.3.11 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 3.3.11
The perimeter of an equilateral triangle is inches. Find the length of each side.
- 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.
Your Turn 3.3.11
Find the length of each side of an equilateral triangle with perimeter inches.
Example 3.3.12
Arianna has inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of inches. How long can she make the two equal sides?
- 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.
Your Turn 3.3.12
A backyard deck is in the shape of an isosceles triangle with a base of feet. The perimeter of the deck is feet. How long is each of the equal sides of the deck?
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 and the length of the bigger base The height, of a trapezoid is the distance between the two bases as shown in Figure 3.3.12.
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 3.3.13.
The formula for the area of a trapezoid is
If we distribute, we get,
Properties of Trapezoids
- A trapezoid has four sides.
- Two of its sides are parallel and two sides are not.
- The area, of a trapezoid is .
Example 3.3.13
Find the area of a trapezoid whose height is 6 inches and whose bases are and inches.
- Answer
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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 the area of the trapezoid. 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 height its area should be greater than that of the trapezoid.
If we draw a rectangle inside the trapezoid that has the same little base and a height its area should be smaller than that of the trapezoid.
The area of the larger rectangle is square inches and the area of the smaller rectangle is square inches. So it makes sense that the area of the trapezoid is between and square inches
Step 7. Answer the question. The area of the trapezoid is 75 square inches.
Your Turn 3.3.13
The height of a trapezoid is yards and the bases are and yards. What is the area?
Example 3.3.14
Find the area of a trapezoid whose height is feet and whose bases are and feet.
- 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.
Your Turn 3.3.14
The height of a trapezoid is centimeters and the bases are and centimeters. What is the area?