3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

Simplify: 1 2 ( 6 h ) . 1 2 ( 6 h ) .
If you missed this problem, review Example 1.122.

Be Prepared 3.13

The length of a rectangle is three less than the width. Let w represent the width. Write an expression for the length of the rectangle.
If you missed this problem, review Example 1.26.

Be Prepared 3.14

Solve: A = 1 2 b h A = 1 2 b h for b when A = 260 A = 260 and h = 52 . h = 52 .
If you missed this problem, review Example 2.61.

Be Prepared 3.15

Simplify: 144 . 144 .
If you missed this problem, review Example 1.111.

Solve Applications Using Properties of Triangles

In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

How To

Solve Geometry Applications.

Step 1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
Step 2. Identify what we are looking for.
Step 3. Label what we are looking for by choosing a variable to represent it.
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 by substituting it back into the equation solved in step 5 and by making sure it makes sense in the context of the problem.
Step 7. Answer the question with a complete sentence.

We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

The plural of the word vertex is vertices. All triangles have three vertices . Triangles are named by their vertices: The triangle in Figure 3.4 is called △ A B C . △ A B C .

A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively.

Figure 3.4 Triangle ABC has vertices A, B, and C. The lengths of the sides are a, b, and c.

The three angles of a triangle are related in a special way. The sum of their measures is 180 ° . 180 ° . Note that we read m ∠ A m ∠ A as “the measure of angle A.” So in △ A B C △ A B C in Figure 3.4,

m ∠ A + m ∠ B + m ∠ C = 180 ° m ∠ A + m ∠ B + m ∠ C = 180 °

Because the perimeter of a figure is the length of its boundary, the perimeter of △ A B C △ A B C is the sum of the lengths of its three sides.

P = a + b + c P = a + b + c

To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a 90 ° 90 ° angle with the base. We will draw △ A B C △ A B C again, and now show the height, h. See Figure 3.5.

A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively. The side b is parallel to the bottom of the page, and it has a dashed line drawn from vertex B to it. This line is marked h and makes a right angle with side b.

Figure 3.5 The formula for the area of △ A B C △ A B C is A = 1 2 b h , A = 1 2 b h , where b is the base and h is the height.

Triangle Properties

For △ A B C △ A B C

Angle measures:

m ∠ A + m ∠ B + m ∠ C = 180 m ∠ A + m ∠ B + m ∠ C = 180

The sum of the measures of the angles of a triangle is 180 ° . 180 ° .

Perimeter:

P = a + b + c P = a + b + c

The perimeter is the sum of the lengths of the sides of the triangle.

Area:

A = 1 2 b h , b = base , h = height A = 1 2 b h , b = base , h = height

The area of a triangle is one-half the base times the height.

Example 3.34

The measures of two angles of a triangle are 55 and 82 degrees. Find the measure of the third angle.

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 measure of the third angle in a triangle
Step 3. Name. Choose a variable to represent it.	Let x = x = the measure of the angle.
Step 4. Translate.
Write the appropriate formula and substitute.	m ∠ A + m ∠ B + m ∠ C = 180 m ∠ A + m ∠ B + m ∠ C = 180
Step 5. Solve the equation.	55 + 82 + x = 180 137 + x = 180 x = 43 55 + 82 + x = 180 137 + x = 180 x = 43
Step 6. Check. 55 + 82 + 43 ≟ 180 180 = 180 ✓ 55 + 82 + 43 ≟ 180 180 = 180 ✓
Step 7. Answer the question.	The measure of the third angle is 43 degrees.

Try It 3.67

The measures of two angles of a triangle are 31 and 128 degrees. Find the measure of the third angle.

Try It 3.68

The measures of two angles of a triangle are 49 and 75 degrees. Find the measure of the third angle.

Example 3.35

The perimeter of a triangular garden is 24 feet. The lengths of two sides are four feet and nine feet. How long is the third side?

Solution

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 = c = the third side.
Step 4. Translate.
Write the appropriate formula and substitute.
Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check. P = a + b + c 24 ≟ 4 + 9 + 11 24 = 24 ✓ P = a + b + c 24 ≟ 4 + 9 + 11 24 = 24 ✓
Step 7. Answer the question.	The third side is 11 feet long.

Try It 3.69

The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 feet. How long is the third side?

Try It 3.70

The lengths of two sides of a triangular window are seven feet and five feet. The perimeter is 18 feet. How long is the third side?

Example 3.36

The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	Area = 90 m 2 = 90 m 2
Step 2. Identify what you are looking for.	height of a triangle
Step 3. Name. Choose a variable to represent it.	Let h = h = the height.
Step 4. Translate.
Write the appropriate formula.
Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check. A = 1 2 b h 90 ≟ 1 2 ⋅ 15 ⋅ 12 90 = 90 ✓ A = 1 2 b h 90 ≟ 1 2 ⋅ 15 ⋅ 12 90 = 90 ✓
Step 7. Answer the question.	The height of the triangle is 12 meters.

Try It 3.71

The area of a triangular painting is 126 square inches. The base is 18 inches. What is the height?

Try It 3.72

A triangular tent door has an area of 15 square feet. The height is five feet. What is the base?

The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A right triangle has one 90 ° 90 ° angle, which we usually mark with a small square in the corner.

Right Triangle

A right triangle has one 90 ° 90 ° angle, which is often marked with a square at the vertex.

Example 3.37

One angle of a right triangle measures 28 ° . 28 ° . What is the measure of the third angle?

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 measure of an angle
Step 3. Name. Choose a variable to represent it.	Let x = x = the measure of an angle.
Step 4. Translate.	m ∠ A + m ∠ B + m ∠ C = 180 m ∠ A + m ∠ B + m ∠ C = 180
Write the appropriate formula and substitute.	x + 90 + 28 = 180 x + 90 + 28 = 180
Step 5. Solve the equation.	x + 118 = 180 x = 62 x + 118 = 180 x = 62
Step 6. Check. 180 ≟ 90 + 28 + 62 180 = 180 ✓ 180 ≟ 90 + 28 + 62 180 = 180 ✓
Step 7. Answer the question.	The measure of the third angle is 62 ° .

Try It 3.73

One angle of a right triangle measures 56 ° . 56 ° . What is the measure of the other small angle?

Try It 3.74

One angle of a right triangle measures 45 ° . 45 ° . What is the measure of the other small angle?

In the examples we have seen so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.

Example 3.38

The measure of one angle of a right triangle is 20 degrees more than the measure of the smallest angle. Find the measures of all three angles.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the measures of all three angles
Step 3. Name. Choose a variable to represent it.	Let a = 1 st a = 1 st angle. a + 20 = 2 nd a + 20 = 2 nd angle 90 = 3 rd 90 = 3 rd angle (the right angle)
Draw the figure and label it with the given information
Step 4. Translate
Write the appropriate formula. Substitute into the formula.
Step 5. Solve the equation.	55 90 third angle
Step 6. Check. 35 + 55 + 90 ≟ 180 180 = 180 ✓ 35 + 55 + 90 ≟ 180 180 = 180 ✓
Step 7. Answer the question.	The three angles measure 35 ° , 55 ° , and 90 ° .

Try It 3.75

The measure of one angle of a right triangle is 50° more than the measure of the smallest angle. Find the measures of all three angles.

Try It 3.76

The measure of one angle of a right triangle is 30° more than the measure of the smallest angle. Find the measures of all three angles.

Use the Pythagorean Theorem

We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem . This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.

Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a 90 ° 90 ° angle, marked with a small square in the corner. The side of the triangle opposite the 90 ° 90 ° angle is called the hypotenuse and each of the other sides are called legs.

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle, a 2 + b 2 = c 2 , a 2 + b 2 = c 2 , where a and b a and b are the lengths of the legs and c c is the length of the hypotenuse.

Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.

The Pythagorean Theorem

In any right triangle, a 2 + b 2 = c 2 . a 2 + b 2 = c 2 .

where a and b are the lengths of the legs, c is the length of the hypotenuse.

To solve exercises that use the Pythagorean Theorem, we will need to find square roots. We have used the notation m m and the definition:

If m = n 2 , m = n 2 , then m = n , m = n , for n ≥ 0 . n ≥ 0 .

For example, we found that 25 25 is 5 because 25 = 5 2 . 25 = 5 2 .

Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.

Example 3.39

Use the Pythagorean Theorem to find the length of the hypotenuse shown below.

Solution

squared plus b squared equals c squared. Below this, we have 3 squared plus 4 squared equals c squared. Step 5. Solve the equation. To the right of this, we have 9 plus 16 equals c squared. We are told to simplify, and hence we have 25 equals c squared. We use the definition of square root to obtain that the square root of 25 equals c. We simplify to obtain that 5 equals c. Step 6. Check. Under this we have 3 squared plus 4 squared equals 5 squared, below which we have 9 plus 16 equals with a question mark over it 25. Below this, 25 equals 25 with a check mark next to it. Step 7. Answer the question: The length of the hypotenuse is 5." data-label="">

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length of the hypotenuse of the triangle
Step 3. Name. Choose a variable to represent it. Label side c on the figure.	Let c = the length of the hypotenuse.
Step 4. Translate.
Write the appropriate formula.	a 2 + b 2 = c 2 a 2 + b 2 = c 2
Substitute.	3 2 + 4 2 = c 2 3 2 + 4 2 = c 2
Step 5. Solve the equation.	9 + 16 = c 2 9 + 16 = c 2
Simplify.	25 = c 2 25 = c 2
Use the definition of square root.	25 = c 25 = c
Simplify.	5 = c 5 = c
Step 6. Check.
Step 7. Answer the question.	The length of the hypotenuse is 5.

Try It 3.77

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

Try It 3.78

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown below.

Example 3.40

Use the Pythagorean Theorem to find the length of the leg shown below.

Solution

squared plus b squared equals c squared. Below this, we have 5 squared plus b squared equals 13 squared. Step 5. Solve the equation. Isolate the variable term. Use the definition of square root. Simplify. To the right of this, we have 25 plus b squared equals 169. Below this, we have b squared equals 144. Below this, b equals the square root of 144. Below this, b equals 12. Step 6. Check. Under this we have 5 squared plus 12 squared equals with a question mark over it 13 squared, below which we have 25 plus 144 equals with a question mark over it 169. Below this, 169 equals 169 with a check mark next to it. Step 7. Answer the question: The length of the leg is 12." data-label="">

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length of the leg of the triangle
Step 3. Name. Choose a variable to represent it.	Let b = the leg of the triangle.
Lable side b.
Step 4. Translate
Write the appropriate formula.	a 2 + b 2 = c 2 a 2 + b 2 = c 2
Substitute.	5 2 + b 2 = 13 2 5 2 + b 2 = 13 2
Step 5. Solve the equation.	25 + b 2 = 169 25 + b 2 = 169
Isolate the variable term.	b 2 = 144 b 2 = 144
Use the definition of square root.	b 2 = 144 b 2 = 144
Simplify.	b = 12 b = 12
Step 6. Check.
Step 7. Answer the question.	The length of the leg is 12.

Try It 3.79

Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

Try It 3.80

Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.

Example 3.41

Kelvin is building a gazebo and wants to brace each corner by placing a 10 ″ 10 ″ piece of wood diagonally as shown above.

If he fastens the wood so that the ends of the brace are the same distance from the corner, what is the length of the legs of the right triangle formed? Approximate to the nearest tenth of an inch.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let x = x = the distance from the corner.
Step 4. Translate Write the appropriate formula and substitute.	a 2 + b 2 = c 2 x 2 + x 2 = 10 2 a 2 + b 2 = c 2 x 2 + x 2 = 10 2
Step 5. Solve the equation. Isolate the variable. Use the definition of square root. Simplify. Approximate to the nearest tenth.	2 x 2 = 100 x 2 = 50 x = 50 x ≈ 7.1 2 x 2 = 100 x 2 = 50 x = 50 x ≈ 7.1
Step 6. Check. a 2 + b 2 = c 2 ( 7.1 ) 2 + ( 7.1 ) 2 ≈ 10 2 Yes. a 2 + b 2 = c 2 ( 7.1 ) 2 + ( 7.1 ) 2 ≈ 10 2 Yes.
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

Try It 3.81

John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. How far up the wall does the ladder reach?

Try It 3.82

Randy wants to attach a 17 foot string of lights to the top of the 15 foot mast of his sailboat, as shown below. How far from the base of the mast should he attach the end of the light string?

Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right ( 90 ° ) ( 90 ° ) angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L, and its adjacent side as the width, W.

The distance around this rectangle is L + W + L + W , L + W + L + W , or 2 L + 2 W . 2 L + 2 W . This is the perimeter , P, of the rectangle.

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

What about the area of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.

A = 6 A = 2 · 3 A = L · W A = 6 A = 2 · 3 A = L · W

The area is the length times the width.

The formula for the area of a rectangle is A = L W . A = L W .

Properties of Rectangles

Rectangles have four sides and four right ( 90 ° ) ( 90 ° ) angles.

The lengths of opposite sides are equal.

The perimeter of a rectangle is the sum of twice the length and twice the width.

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

The area of a rectangle is the product of the length and the width.

A = L · W A = L · W

Example 3.42

The length of a rectangle is 32 meters and the width is 20 meters. What is the perimeter?

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. P ≟ 104 20 + 32 + 20 + 32 ≟ 104 104 = 104 ✓ P ≟ 104 20 + 32 + 20 + 32 ≟ 104 104 = 104 ✓
Step 7. Answer the question.	The perimeter of the rectangle is 104 meters.

Try It 3.83

The length of a rectangle is 120 yards and the width is 50 yards. What is the perimeter?

Try It 3.84

The length of a rectangle is 62 feet and the width is 48 feet. What is the perimeter?

Example 3.43

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

Solution

the width of a rectangular room. Step 3. Name. Choose a variable to represent it. To the right of this, we have let W equal the width. Step 4. Translate. Write the appropriate formula. To the right of this, we have A equals LW. We are told to substitute. To the right of this, we have 168 equals 14W. Step 5. Solve the equation. To the right of this, we have 168/14 equals 14W/14. Below this, we have 12 equals W. Step 6. Check. Under this we have a rectangle with width 12 ft and length 14 ft. Below that, we have A equals LW. Below this, we have 168 equals with a question mark over it 14 times 12. Below this, 168 equals 168. Step 7. Answer the question: The width of the room is 12 feet." data-label="">

Step 1. Read the problem. Draw the figure and label it with the given information.
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 = the width.
Step 4. Translate.
Write the appropriate formula.	A = L W A = L W
Substitute.	168 = 14 W 168 = 14 W
Step 5. Solve the equation.	168 14 = 14 W 14 168 14 = 14 W 14 12 = W 12 = W
Step 6. Check. A = L W 168 ≟ 14 ⋅ 12 168 = 168 ✓ A = L W 168 ≟ 14 ⋅ 12 168 = 168 ✓
Step 7. Answer the question.	The width of the room is 12 feet.

Try It 3.85

The area of a rectangle is 598 square feet. The length is 23 feet. What is the width?

Try It 3.86

The width of a rectangle is 21 meters. The area is 609 square meters. What is the length?

Example 3.44

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.
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. P = 50 15 + 10 + 15 + 10 ≟ 50 50 = 50 ✓ P = 50 15 + 10 + 15 + 10 ≟ 50 50 = 50 ✓
Step 7. Answer the question.	The length is 15 inches.

Try It 3.87

Find the length of a rectangle with: perimeter 80 and width 25.

Try It 3.88

Find the length of a rectangle with: perimeter 30 and width 6.

We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

Example 3.45

The width of a rectangle is two feet less than the length. The perimeter is 52 feet. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width of a rectangle
Step 3. Name. Choose a variable to represent it. Since the width is defined in terms of the length, we let L = length. The width is two feet less than the length, so we let L − 2 = width.	P = 52 P = 52 ft
Step 4. Translate.
Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information.	P = 2 L + 2 W P = 2 L + 2 W
Substitute in the given information.	52 = 2 L + 2 ( L − 2 ) 52 = 2 L + 2 ( L − 2 )
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 feet.
Now we need to find the width.	The width is L − 2 L − 2 . The width is 12 feet.
Step 6. Check. Since 14 + 12 + 14 + 12 = 52 14 + 12 + 14 + 12 = 52 , this works!
Step 7. Answer the question.	The length is 14 feet and the width is 12 feet.

Try It 3.89

The width of a rectangle is seven meters less than the length. The perimeter is 58 meters. Find the length and width.

Try It 3.90

The length of a rectangle is eight feet more than the width. The perimeter is 60 feet. Find the length and width.

Example 3.46

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 the width
Step 3. Name. Choose a variable to represent the width.
The length is four more than twice the width.
Step 4. Translate
Write the appropriate formula.
Substitute in the given information.
Step 5. Solve the equation.	12 The length is 12 cm.
Step 6. Check. P = 2 L + 2 W 32 ≟ 2 ⋅ 12 + 2 ⋅ 4 32 = 32 ✓ P = 2 L + 2 W 32 ≟ 2 ⋅ 12 + 2 ⋅ 4 32 = 32 ✓
Step 7. Answer the question.	The length is 12 cm and the width is 4 cm.

Try It 3.91

The length of a rectangle is eight more than twice the width. The perimeter is 64. Find the length and width.

Try It 3.92

The width of a rectangle is six less than twice the length. The perimeter is 18. Find the length and width.

Example 3.47

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.	P = 150 P = 150 ft
Step 2. Identify what you are looking for.	the length and the width of the pool
Step 3. Name. Choose a variable to represent the width. The length is 15 feet more than the width.
Step 4. Translate
Write the appropriate formula.
Substitute.
Step 5. Solve the equation.
Step 6. Check. P = 2 L + 2 W 150 ≟ 2 ( 45 ) + 2 ( 30 ) 150 = 150 ✓ P = 2 L + 2 W 150 ≟ 2 ( 45 ) + 2 ( 30 ) 150 = 150 ✓
Step 7. Answer the question.	The length of the pool is 45 feet and the width is 30 feet.

Try It 3.93

The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.

Try It 3.94

The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.

Section 3.4 Exercises

Practice Makes Perfect

Solving Applications Using Triangle Properties

In the following exercises, solve using triangle properties.

The measures of two angles of a triangle are 26 and 98 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 61 and 84 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 105 and 31 degrees. Find the measure of the third angle.

The measures of two angles of a triangle are 47 and 72 degrees. Find the measure of the third angle.

The perimeter of a triangular pool is 36 yards. The lengths of two sides are 10 yards and 15 yards. How long is the third side?

A triangular courtyard has perimeter 120 meters. The lengths of two sides are 30 meters and 50 meters. How long is the third side?

If a triangle has sides 6 feet and 9 feet and the perimeter is 23 feet, how long is the third side?

If a triangle has sides 14 centimeters and 18 centimeters and the perimeter is 49 centimeters, how long is the third side?

A triangular flag has base one foot and height 1.5 foot. What is its area?

A triangular window has base eight feet and height six feet. What is its area?

What is the base of a triangle with area 207 square inches and height 18 inches?

What is the height of a triangle with area 893 square inches and base 38 inches?

One angle of a right triangle measures 33 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 51 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 22.5 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 36.5 degrees. What is the measure of the other small angle?

The perimeter of a triangle is 39 feet. One side of the triangle is one foot longer than the second side. The third side is two feet longer than the second side. Find the length of each side.

The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.

One side of a triangle is twice the shortest side. The third side is five feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.

One side of a triangle is three times the shortest side. The third side is three feet more than the shortest side. The perimeter is 13 feet. Find the lengths of all three sides.

The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

The measure of the smallest angle of a right triangle is 20° less than the measure of the next larger angle. Find the measures of all three angles.

The angles in a triangle are such that one angle is twice the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

The angles in a triangle are such that one angle is 20° more than the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.