Radical Expressions and Equations

The Pythagorean Theorem

Objective: To solve problems using the Pythagorean Theorem. To identify right triangles.

Objectives

• I can find the length of a hypotenuse.

• I can find the length of a leg.

• I can identify right triangles.

Vocabulary

• There are special names for the sides of a right triangle.

• The side opposite the right angle is the hypotenuse. It is the longest side.

• Each of the sides forming the right angle is a leg.

• The Pythagorean Theorem, named after the Greek mathematician Pythagoras, relates the lengths of the legs and the length of the hypotenuse.

Vocabulary

• The lengths of the sides of a right triangle have a special relationship. If you know the lengths of any two of the sides, you can find the length of the third side.

• The Pythagorean Theorem:

• Words: If any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

• Algebra: 𝑎2 + 𝑏2 = 𝑐2

• You can use the Pythagorean Theorem to find the length of a right triangle’s hypotenuse given the lengths of its legs. Using the Pythagorean Theorem to solve for a side length involves finding a principal square root, because side lengths are always positive.

Finding the Length of a Hypotenuse

• The picture is a square with 6 inch sides. What is the length of the hypotenuse of the right triangle shown?

• What is the length of the hypotenuse of a right triangle with legs of lengths 9 cm and 12 cm?

Practice

Use the triangle to find the hypotenuse. Round to the nearest tenth.1. a = 3, b = 4

2. a = 0.3, b = 0.4

3. a = 8, b = 15

4. a = 1.1, b = 6

5. a = 9, b = 40

6. a = 4, b = 7.5

Vocabulary

• You can also use the Pythagorean Theorem to find the length of a leg of a right triangle.

Finding the Length of a Leg

• What is the side length b when one leg is 5 cm and the hypotenuse is 13 cm?

• What is the side length a when one leg is 12 and the hypotenuse is 15.

Practice

Use the triangle to find the missing length. Round to the nearest tenth.1. a = 6, c = 10

2. b = 1, c = 5

4

3. a = 5, c = 13

4. a = 1, b = 5

3

5. b = 6, c = 7.5

6. b = 3.5, c = 3.7

7. a = 8, c = 17

8. b = 2.4, c = 7.4

9. a = 0.9, c = 4.1

Vocabulary

• An if-then statement such as “If an animal is a horse, then it has four legs” is called a conditional.

• Conditionals have two parts. The part following if is the hypothesis. The part following then is the conclusion.

• The converse of a conditional switches the hypothesis and the conclusion. Sometimes the converse of a true conditional is not true.

• You can write the Pythagorean Theorem as a conditional: “If a triangle is a right triangle with legs of length a and b and hypotenuse of length c, then 𝑎2 + 𝑏2 = 𝑐2.” The converse of the Pythagorean Theorem is always true.

Vocabulary

• The converse of the Pythagorean Theorem:

• If a triangle has sides of lengths a, b, and c, and 𝑎2 + 𝑏2 = 𝑐2, then the triangle is a right triangle with hypotenuse of length c.

• You can use the Pythagorean Theorem and its converse to determine whether a triangle is a right triangle. If the side lengths satisfy the equations 𝑎2 + 𝑏2 = 𝑐2, then the triangle is a right triangle. If they do not, then it is not a right triangle.

Identifying Right Triangles

• Which set of lengths could be the side lengths of a right triangle?1. 6 in, 24 in, 25 in

2. 4 m, 8 m, 10 m

3. 10 in, 24 in, 26 in

4. 8 ft, 15 ft, 16 ft

• Could the lengths of 20 mm, 47 mm, and 52 mm be the side lengths of a right triangle?

Practice

Determine whether the given lengths can be side lengths of a right triangle.1. 15 ft, 36 ft, 39 ft

2. 16 cm, 63 cm, 65 cm

3. 12 m, 60 m, 61 m

4. 14 in, 48 in, 50 in

5. 13 in, 35 in, 38 in

6. 16 yd, 30 yd, 34 yd

Practice

• A jogger goes half a mile north and then turns west. If the jogger finishes 1.3 miles from the starting point, how far west did the jogger go?

• A construction worker is cutting along the diagonal of a rectangular board 15 feet long and 8 feet wide. What will be the length of the cut?

Simplifying Radicals

Objective: To simplify radicals involving products and quotients.

Objectives

• I can remove perfect-square factors.

• I can remove variable factors.

• I can multiply two radical expressions.

• I can write a radical expression.

• I can simplify fractions within radicals.

• I can rationalize denominators.

Vocabulary

• A radical expression, such as 2 3 𝑜𝑟 𝑥 + 3, is an expression that contains a radical.

• A radical expression is simplified if the following statements are true.1. The radicand has no perfect square factors other than 1.

2. The radicand contains no fractions.

3. No radicals appear in the denominator of a fraction.

Simplified Not Simplified

3 5, 9 𝑥, 2

4 3 12, 𝑥

2,

5

7

Vocabulary

• You can simplify radical expressions using multiplication and division properties of square roots.

• Multiplication Property of Square Roots:• Algebra: For a ≥ 0 and b ≥ 0, 𝑎𝑏 = 𝑎 ∙ 𝑏

• Example: 48 = 16 ∙ 3 = 4 3

• You can use the Multiplication Property of Square Roots to simplify radicals to removing perfect-square factors from the radicand.

Removing Perfect – Square Factors

What is the simplified form?1. 160

2. 72

3. 98

4. 225

5. 99

Practice

Simplify each radical expression.1. 128

2. − 60

3. −4 117

4. 5 700

5. −5 175

6. 80

Vocabulary

• Sometimes you can simplify radical expressions that contain variables.

• A variable with an even exponent is a perfect square.

• A variable with an odd exponent is the product of a perfect square and the variable. For example 𝑛3 = 𝑛2 ∙ 𝑛, 𝑠𝑜 𝑛3 = 𝑛2 ∙ 𝑛.

Removing Variable Factors

What is the simplified form?

1. 54𝑛7

2. −𝑚 80𝑚9

3. 16𝑏5

4. 192𝑠2

5. 50𝑡5

Practice

Simplify each radical expression.

1. 3 18𝑎2

2. −21 27𝑥9

3. 3 150𝑏8

4. −2 243𝑦3

5. 144𝑥5

6. 20𝑎2𝑏3

Vocabulary

• You can use the Multiplication Property of Square Roots to write 𝑎 ∙ 𝑏 = 𝑎𝑏.

Multiplying Two Radical Expressions

What is the simplified form?

1. 2 7𝑡 ∙ 3 14𝑡2

2. 3 6 ∙ 18

3. 2𝑎 ∙ 9𝑎3

4. 7 5𝑥 ∙ 3 20𝑥5

5. 3 5𝑚 ∙ 41

5𝑚3

6. 8 ∙ 32

Practice

Simplify each product.

1. 5 6 ∙1

6216

2. −5 21 ∙ −3 42

3. 18𝑛 ∙ 98𝑛3

4. 3 5𝑐 ∙ 7 15𝑐2

5. 2𝑦 ∙ 128𝑦5

6. −6 15𝑠3 ∙ 2 75

Writing a Radical Expression

• A rectangular door in a museum is three times as tall as it is wide. What is the simplified expression for the maximum length of a painting that fits through the door?

• A door’s height is four times its width w. What is the maximum length of a painting that fits through the door?

Practice

• Students are building rectangular wooden frames for the set of a school play. The height of a frame is 6 times the width w. Each frame has a brace that connects two opposite corners of the frame. What is the simplified expression for the length of the brace?

• A park is shaped like a rectangle with a length 5 times its width w. What is the simplified expression for the distance between opposite corners of the park?

Vocabulary

• Division Property of Square Roots:

• Algebra: for a ≥ 0 and b > 0, 𝑎

𝑏=

𝑎

𝑏.

• Example: 36

49=

36

49=

6

7

• When a radicand has a denominator that is a perfect square, it is easier to apply the Division Property of Square Roots first and then simplify the numerator and denominator of the result. When the denominator of a radicand is not a perfect square, it may be easier to simplify the fraction first.

Simplifying Fractions Within Radicals

What is the simplified form of each radical expression?

1.64

49

2.8𝑥3

50𝑥

3.144

9

4.36𝑎

4𝑎3

5.25𝑦3

𝑧2

6.15𝑥

𝑥3

Practice

Simplify each radical expression.

1.16

25

2. 7 ∙6

32

3. −4 ∙100

729

4.3𝑥3

64𝑥2

5. −5 ∙162𝑡3

2𝑡

6. 11 ∙49𝑎5

4𝑎3

Vocabulary

• When a radicand in a denominator is not a perfect square, you may need to rationalize the denominator to remove the radical.

• To do this, multiply the numerator and the denominator by the same radical expression.

• Choose an expression that make the radicand in the denominator a perfect square.

• It may be helpful to start by simplifying the original radical in the denominator.

Rationalizing Denominators

What is the simplified form of each expression?

1.3

7

2.7

8𝑛

3.2

3

4.5

18𝑚

5.7𝑠

3

6.6

2𝑛

Practice

Simplifiy each radical expression.

1.1

11

2.5

8𝑥

3.3 6

15

4.22

11

5.2 24

48𝑡4

6.8 7𝑠

28𝑠3

Operations With Radical Expressions

Objective: To simplify sums and differences of radical expressions. To simplify products and quotients of radical expressions.

Objectives

• I can combine like radicals.

• I can simplify to combine like radicals.

• I can multiply radical expressions.

• I can rationalize a denominator using conjugates.

• I can solve a proportion involving radicals.

Vocabulary

• You can use properties of real numbers to perform operations with radical expressions.

• For example, you can use the distributive property to simplify sums or difference of radical expressions by combining like radicals.

• Like radicals, such as 3 5 𝑎𝑛𝑑 7 5, have the same radicand.

• Unlike radicals, such as 4 3 𝑎𝑛𝑑 − 2 2, have different radicands.

Combining Like Radicals

What is the simplified form of each expression?1. 6 11 + 9 11

2. 3 − 5 3

3. 7 2 − 8 2

4. 5 5 + 2 5

5. 4 3 + 3

Practice

Simplify each sum or difference.1. 5 + 6 5

2. 12 5 − 3 5

3. 7 3 + 3

4. 4 2 − 7 2

5. 2 10 + 3 10

Vocabulary

• You may need to simplify radical expressions first to determine if they can be added or subtracted by combining like radicals.

Simplifying to Combine Like Radicals

What is the simplified form?1. 5 3 − 12

2. 4 7 + 2 28

3. 5 32 − 4 18

4. 3 6 − 24

5. 3 7 − 63

Practice

Simplify each sum or difference.1. 4 128 + 5 18

2. 3 45 − 8 20

3. 28 − 5 7

4. −6 10 + 5 90

5. 3 3 − 2 12

6. −1

25 + 2 125

7. 5 8 + 2 72

Vocabulary

• When simplifying a product like 10( 6 + 3), you can use the Distributive Property to multiply 10 times 6 and 10 times 3.

• If both factors in the product have two terms, as in 6 − 2 3 6 + 3 , you can use FOIL to multiply just as you do when multiplying binomials.

Multiplying Radical Expressions

What is the simplified form of each expression?

1. 10 6 + 3

2. 6 − 2 3 6 + 3

3. 2 6 + 5

4. 11 − 22

5. 6 − 2 3 4 3 + 3 6

6. 6 2 + 3

7. 5 15 − 3

Practice

Simplify each product.

1. 3 7 1 − 7

2. − 12 4 − 2 3

3. 5 11 3 − 3 2

4. 3 11 + 72

5. 2 + 10 2 − 10

6. 6 + 3 2 − 2

7. 5 2 − 2 32

Vocabulary

• Conjugates are the sum and difference of the same two terms.

• For example, 7 + 3 𝑎𝑛𝑑 7 − 3 are conjugates.

• The product of conjugates is a difference of squares.

7 + 3 7 − 3 = 72

− 32

= 7 − 3 = 4

• You can use conjugates to simplify a quotient whose denominator is a sum or difference of radicals.

Rationalizing a Denominator Using Conjugates

What is the simplified form?

1.10

7− 2

2.−3

10+ 5

3.7 5

3+ 2

4.6

7+2

5.5

2−1

Practice

Simplify each quotient.

1.3

7− 3

2.−2

6+ 11

3.5

2− 5

4.−1

2−2 3

5.7

5+ 13

Vocabulary

• Golden rectangles appear frequently in nature and art. The ratio of the length to the width of a golden rectangle is 1 + 5 : 2.

Solving a Proportion Involving Radicals

• Fiddlehead ferns naturally grow in spirals that fit into golden rectangles. What is the width w of the ferns when the rectangle is 4 cm?

• A golden rectangle is 12 inches long. What is the width of the rectangle? Write your answer in simplified form. Round to the nearest tenth of an inch.

Practice

• A shell fits into a golden rectangle with a length of 8 inches. What is the shell’s width? Write your answer in simplified radical form and rounded to the nearest tenth of an inch.

• A room is approximately shaped like a golden rectangle. Its length is 23 feet. What is the room’s width? Write your answer in simplified radical form and rounded to the nearest tenth of a foot.

Solving Radical Equations

Objective: To solve equations containing radicals. To identify extraneous solutions.

Objectives

• I can solve by isolating the radical.

• I can use a radical equation.

• I can solve with radical expressions on both sides.

• I can identify extraneous solutions.

• I can identify equations with no solutions.

Vocabulary

• A radical equation is an equation that has a variable in a radicand. Examples include 𝑥 − 5 = 3 𝑎𝑛𝑑 𝑥 − 2 = 1.

• To solve a radical equation, get the radical by itself on one side of the equation. Then square both sides. The expression under the radical must be nonnegative.

• You can solve some radical equations by squaring each side of the equation and testing the solutions.

Solving by Isolating the Radical

What is the solution?1. 𝑥 + 7 = 16

2. 𝑥 − 5 = −2

3. 3𝑥 + 10 = 16

4. 𝑥 + 3 = 5

5. 𝑡 + 2 = 9

6. 𝑧 − 1 = 5

Practice

Solve each radical equation.1. 𝑛 − 3 = 6

2. 2𝑏 + 4 = 8

3. 3 − 𝑡 = −2

4. 3𝑎 + 1 = 7

5. 10𝑏 + 6 = 6

6. 1 = −2𝑣 − 3

7. 𝑥 − 3 = 4

Using a Radical Equation

• The time t in seconds it takes for a pendulum of a clock to complete a full swing is

approximated by the equation 𝑡 = 2 ∙𝑙

3.3, where l is the length of the

pendulum, in feet. If the pendulum of a clock completes a full swing in 3 seconds, what is the length of the pendulum? Round to the nearest tenth of a foot.

• How long is a pendulum if each swing takes 1 second?

Practice

• You are making a tire swing for a playground. The time t in seconds for the tire to

make one swing is given by 𝑡 = 2 ∙𝑙

3.3, where l is the length of the swing in feet.

You want one swing to take 2.5 s. How many feet long should the swing be?

• The length s of one edge of a cube is given by 𝑠 =𝐴

6, where A represents the

cube’s surface area. Suppose a cube has an edge length of 9 cm. What is its surface area? Round to the nearest hundredth.

Solving With Radical Expressions on Both Sides

What is the solution?1. 5𝑡 − 11 = 𝑡 + 5

2. 7𝑥 − 4 = 5𝑥 + 10

3. 𝑟 + 5 = 2 𝑟 − 1

4. 2𝑥 − 1 = 𝑥

5. 𝑥 − 3 = 𝑥 + 5

Practice

Solve each radical equation.1. 3𝑥 + 1 = 5𝑥 − 8

2. 2𝑦 = 9 − 𝑦

3. 7𝑣 − 4 = 5𝑣 + 10

4. 𝑠 + 10 = 6 − 𝑠

5. 𝑛 + 5 = 5𝑛 − 11

6. 3𝑚 + 1 = 7𝑚 − 9

Vocabulary

• When you solve an equation by squaring each side, you create a new equation. The new equation may have solution that do not satisfy the original equation.

• In the example above, –3 does not satisfy the original equation. It is an extraneous solution. An extraneous solution is an apparent solution that does not satisfy the original equation. Always substitute each apparent solution into the original equation to check for extraneous solutions.

Original Equation Square each side New Equation Apparent Solutions

𝑥 = 3 𝑥2 = 32 𝑥2 = 9 3, −3

Identifying Extraneous Solutions

What is the solution?1. 𝑛 = 𝑛 + 12

2. −𝑦 = 𝑦 + 6

3. 2𝑥 − 1 = 𝑥

4. 𝑠 = 𝑠 + 2

5. 7𝑦 + 18 = 𝑦

Practice

Tell which solutions, if any are extraneous for each equation.1. −𝑧 −𝑧 + 6; 𝑧 = −3, 𝑧 = 2

2. 12 − 𝑛 = 𝑛; 𝑛 = −4, 𝑛 = 3

3. 𝑦 = 2𝑦; 𝑦 = 0, 𝑦 = 2

4. 2𝑎 = 4𝑎 + 3; 𝑎 =3

2, 𝑎 =

1

2

5. 𝑥 = 28 − 3𝑥; 𝑥 = 4, 𝑥 = −7

6. −𝑡 = −6𝑡 − 5; 𝑡 = −5, 𝑡 = −1

Vocabulary

• Sometimes you get only extraneous solutions after squaring each side of an equation. In that case, the original equation has not solution.

Identifying Equations With No Solutions

What is the solution?

1. 3𝑦 + 8 = 2

2. 6 − 2𝑥 = 10

3. 𝑥 = −5

4. 5𝑥 + 10 = 5

5. 3 − 4𝑎 + 1 = 12

Practice

Solve each radical equation. Check your solution. If there is no solution, write no solution.

1. 𝑥 = 2𝑥 + 3

2. 𝑛 = 4𝑛 + 5

3. 3𝑏 = −3

4. 2𝑦 = 5𝑦 + 6

5. −2 2𝑟 + 5 = 6

6. 𝑑 + 12 = 𝑑

Graphing Square Root Functions

Objective: To graph square root functions. To translate graphs of square root functions.

Objectives

• I can find the domain of a square root function.

• I can graph a square root function.

• I can graph a vertical translation.

• I can graph a horizontal translation.

Vocabulary

• A square root function is a function containing a square toot with the independent variable in the radicand. The parent square root function is 𝑦 = 𝑥.

• The table and graph show the parent square root function.

• You can graph a square root function by plotting points or using a translation of the parent square root function.

• For real numbers, the value of the radicand cannot be negative. So the domain of a square root is limited to values of x for which the radicand is greater than or equal to 0.

X Y

0 0

1 1

2 1.4

4 2

9 3

Finding the Domain of a Square Root Function

What is the domain of the function?1. 𝑦 = 2 3𝑥 − 9

2. 𝑦 = −2𝑥 + 5

3. 𝑦 = 𝑥 + 3

4. 𝑦 = 𝑥 − 7

5. 𝑦 = 𝑥 + 2

6. 𝑦 =1

2𝑥

Practice

Find the domain of each function.

1. 𝑦 = 3𝑥

3

2. 𝑦 = 2.7 𝑥 + 2 + 11

3. 𝑦 = 4𝑥 − 13

4. 𝑦 =4

718 − 𝑥

5. 𝑦 = 3𝑥 + 9 − 6

6. 𝑦 = 3(𝑥 − 4)

Graphing a Square Root Function

• Graph the function 𝐼 =1

5𝑃,

which gives the current I in amperes for a certain circuit with P watts of power. When will the current exceed 2 amperes?

• When will the current exceed 1.5 amperes?

Graphing a Square Root Function

Graph each function.1. 𝑦 = 2 𝑥

2. 𝑦 = 𝑥 − 6

3. 𝑦 = 4𝑥 − 8

4. 𝑦 = −3 𝑥

5. 𝑦 =𝑥

2

Practice

Make a table of values and graph each function.1. 𝑦 = 2𝑥

2. 𝑦 = 3𝑥

3. 𝑦 =1

3𝑥

4. 𝑦 = 4 𝑥

5. 𝑦 = 3 𝑥

6. 𝑦 = 2 𝑥 − 3

Vocabulary

• For any positive number k, graphing 𝑦 = 𝑥 + 𝑘 translates the graph of y = 𝑥up k units.

• Graphing 𝑦 = 𝑥 − 𝑘 translates the graph of 𝑦 = 𝑥 down k units.

• For any positive number h, graphing 𝑦 = 𝑥 + ℎ translates the graph of 𝑦 = 𝑥to the left h units.

• Graphing 𝑦 = 𝑥 − ℎ translates the graph of 𝑦 = 𝑥 to the right h units.

Graphing a Vertical Translation

What is the graph?1. 𝑦 = 𝑥 + 2

2. 𝑦 = 𝑥 − 3

3. 𝑦 = 𝑥 − 6

4. 𝑦 = 𝑥 + 4

Practice

Graph each function by translating the graph of 𝑦 = 𝑥.1. 𝑦 = 𝑥 + 5

2. 𝑦 = 𝑥 + 1

3. 𝑦 = 𝑥 − 5

4. 𝑦 = 𝑥 − 1

5. 𝑦 = 𝑥 + 4

6. 𝑦 = 𝑥 − 2

Graphing a Horizontal Translation

What is the graph?1. 𝑦 = 𝑥 + 3

2. 𝑦 = 𝑥 − 3

3. 𝑦 = 𝑥 − 2

4. 𝑦 = 𝑥 + 4

Practice

Graph each function by translating the graph of 𝑦 = 𝑥.1. 𝑦 = 𝑥 + 2

2. 𝑦 = 𝑥 − 5

3. 𝑦 = 𝑥 − 4

4. 𝑦 = 𝑥 + 1

5. 𝑦 = 𝑥 − 1

6. 𝑦 = 𝑥 + 5

Trigonometric Ratios

Objective: To find and use trigonometric ratios.

Objectives

• I can find trigonometric ratios.

• I can find a trigonometric ratio.

• I can find a missing side length.

• I can find the measures of angles.

• I can use an angle of elevation or depression.

Vocabulary

• Ratios of the side lengths of right triangle are called trigonometric ratios.

• You can use the sine, cosine, and tangent ratios to find the measurements of sides and angles of right triangles.

Name Written

Definition

Sine of ∠A Sin A length of leg opposite ∠A

length of hypotenuse

Cosine of ∠A Cos A length of leg adjacent to ∠A

length of hypotenuse

Tangent of ∠A Tan A length of leg opposite ∠A

length of leg adjacent to ∠A

Finding Trigonometric Ratios

What are the sin A, cos A, and tan A for the triangle shown?

Practice

• What are the sin A, cos A, and tan A for the triangle shown?

• What are the sin B, cos B, and tan B for the triangle shown?

• What are the sin E, cos E, and tan E for the triangle shown?

• What are the sin F, cos F, and tan F for the triangle shown?

Vocabulary

• You can also use a calculator to find trigonometric ratios.

• In this unit, use Degree mode when finding trigonometric ratios.

• That allows you to enter angles in degrees.

• Go to MODE, scroll down and select DEGREE, then hit CLEAR to exit.

Finding a Trigonometric Ratio

What is the value of each rounded to the nearest ten-thousandth?1. Cos 55°

2. Sin 80°

3. Tan 45°

4. Cos 15°

5. Sin 9°

Practice

Find the value of each expression. Round to the nearest ten-thousandth.

1. Sin 10°

2. Cos 85°

3. Sin 70°

4. Sin 71°

5. Tan 25°

6. Tan 12°

7. Cos 22°

8. Tan 30°

Vocabulary

• You can use trigonometry to find missing lengths in a right triangle when you know the length of one side and the measure of an acute angle.

Finding a Missing Side Length

To the nearest tenth, what is the value of x in the triangle? What is the value of y in the triangle?

Practice

Finding the value of x to the nearest tenth.

14

x

67°

x

x

x

x

x

14

48°

12

32°

25

41°56

32°

36°

29

Vocabulary

• If you know the lengths of two sides of a right triangle, you can find a trigonometric ratio for each acute angle of the triangle.

• If you know a trigonometric ratio for an angle, you can use the inverse of the trigonometric ratio to find the measure of the angle.

• Use the sin−1, cos−1, or tan−1 feature on your calculator.

• To use the inverses you will need to hit the 2nd button then sin, cos, or tan.

Finding the Measures of Angles

What is the measure of each angle in the triangle?

Practice

Find the value of x to the nearest degree.

5

10x°

3

x°

3 2

5

6.7

x°

Vocabulary

• You can use trigonometric ratios to measure some distances indirectly.

• To measure such distances it is often convenient to use an angle of elevation or an angle of depression.

• An angle of elevation is an angle from the horizontal up to a line of sight.

• An angle of depression is an angle from the horizontal down to a line of sight.

• When you solve real world problems using trigonometric ratios, you often need to round your answers. The problem may tell you to round. Otherwise, round your answers to the precision of the measurements used in the problem.

Using an Angle of Elevation or Depression

• Suppose you are waiting in line for a ride. You see your friend at the top of the ride. How far are you from the base of the ride?

• After you move forward in the line, the angle of elevation to the top of the ride becomes 50°. How far are you from the base of the ride now?

150 feet

20°

20°

x

Practice

• From an observation point 20 feet from the base of a geyser, the angle of elevation to the top is 50. How tall is the geyser?

• A wheelchair ramp is being planned for a new building. The ramp will rise a total of 2.5 feet and form a 3 angle with the ground. How far from the base of the building should the wheelchair ramp start?