summerbailey.weebly.com€¦ · Web vieweither SOH-CAH-TOA story, song, or word problem where you solve using sine, cosine or tangent: Groups . Day 12: Review /Reflect. for Chapter

Unit PlanChapter 7: Right Triangles and Trigonometry

Includes:

7.1- Apply the Pythagorean Theorem7.2- Converse to the Pythagorean Theorem7.4- Special Right Triangles7.5-Apply Tangent7.6- Apply Sine and Cosine7.7- Solve Right triangles

Book: Title: Larson GeometryAuthors: Larson, Boswell, Kanold, Stiff2011 by Houghton Mifflin Harcourt Publishing Company

Standards:

G.SRT.4 Prove theorems about triangles.

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

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Day-to-Day Plan:

Day 1: Apply the Pythagorean Theorem Pythagorean Stack Worksheet/Project

Day 2: Right Triangle Around Us Project: GroupsUse the Converse of the Pythagorean Theorem

Day 3: Present Right Triangle Around Us Project: ReflectionSpecial Right Triangles

Day 4: Review: Kahoot quizDay 5: Quiz over 7.1, 7.2, 7.4Day 6: Introduce with Leaning Tower of Pisa

Apply the Tangent RatioDay 7: Apply the Sine and Cosine Ratio Day 8: Article Example

Solve Right trianglesDay 9: Quiz over 7.5, 7.6, 7.7

Article ProjectDay 10: Present Articles to class/Mini PaperDay 11: SOH-CAH-TOA: story, song.

Create either SOH-CAH-TOA story, song, or word problem where you solve using sine, cosine or tangent: Groups

Day 12: Review/Reflect for Chapter Test: On whiteboardsDay 13: Test over Right Triangles and Trigonometry Chapter

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7.1 Applying the Pythagorean TheoremGrade Level: 10th

Subject(s) Area: Geometry

Materials Needed: Pythagorean Theorem Reading Packet, Discussion Chart sheet,

pencil, whiteboard, markers, tape measure (project)

Standards:

G.SRT.4 Prove theorems about triangles.



Objectives: Students will recite the Pythagorean Theorem with no difficultly

correctly within 30 seconds. Students will be able to apply the Pythagorean Theorem to any right

triangle given two side lengths.

Learning Activities:Apply the Pythagorean Theorem

History: Pythagoras Identity the legs and hypotenuse of a right triangle, the acute angles,

the label of the sides, the measure of angle C. Math is Fun: https://www.mathsisfun.com/geometry/triangles-

interactive.html State the Pythagorean Theorem:

o In a right triangle, the square of the length of the Hypotenuse is equal to the sum of the squares of the lengths of the legs.

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https://www.mathsisfun.com/geometry/triangles-interactive.html


o Have students determine Pythagorean Theorem from the definition.

c2=a2+b2

Prove the Pythagorean Theorem: Square Diagram with small diamond inside of c by c. 4 triangles

making a bigger triangle. (a+b)2=4(.5ab)+c2

Examples of Pythagorean Theorem: On board, individual, groupso Find hypotenuse with legs 8,6. Do for students showing how to

apply the Pythagorean Theorem. 10o Find leg with leg 3 and hypotenuse 5. Have students walk me

through this problem. 4o Find hypotenuse with legs 4 and 6. Have students do this in

work then either go over with them or have a student come to board. 2sqrt(13)

o A 16 foot ladder rests against the side of the house, and the base of the ladder is 4 feet away. Approximately how high above the ground is the top of the ladder? Draw house and ladder, have students tell you which measurement goes where, how to set up equation and how to solve it.

o Find area of isosceles triangle with sides 10, 13, 13. Review what an isosceles triangle is (text context). Draw and label the triangle, discuss how to find the area of the triangle. What is the equation for area of a triangle? .5bh. What is the base? What is the height? How do I find the height? Use Pythagorean Theorem.

Pythagorean Tripleo A series of three positive integers a, b, c that satisfy the

equation c2=a2+b2. 3,4,5 ; 5,12,13 ; 6,8,10 ; 9,12,15 .

Homework: Pythagorean Stack Worksheet 2-32

Think of an example at where you would use the Pythagorean Theorem in everyday life? Building rafters

Project: Go around the school with a tape measure to find a right angle (stair case), measure two of the sides, find the length of the third side using

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the Pythagorean Theorem. Then measure the third side (if possible) to see if your answer is correct. Come up with a word problem scenario with your right triangle in the school. Such as: (Stair case) Billy want to slide down the rail of the stair case, how far will he slide if the triangle is 12 ft high and 17 feet long?

Assessment: Formal Assessment: Quiz or/and Test:

o Quiz taken in four dayso Test taken at end of chapter

Informal Assessment: Project, Board Worko Project due in two days

Reflection: What did I do well? If I were to teach this lesson again, what would I keep the same? If I were to teach this lesson again, what would I change? This is completed after your lesson is taught.

The questions listed below each subheading are only guidelines when creating your lesson.

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Right Triangle Around Us RubricStudents will be in groups finding a right triangle in the school or classroom that has a “mystery” side to find the length of. Measure two sides of the right triangle, use the Pythagorean Theorem to find the missing third side, draw a right triangle that pertains to your selected right triangle (label everything you can), write out the equation to find the missing side (show all your work) and present your finding with your group in front of the class.

5 4 3 2 1Quality of Right Triangle Drawing

Drawing was labeled for every part.

Drawing was labeled excluding 1-2 obvious labels.

Drawing included necessary labels to understand drawing.

Drawing excluded 1-2 necessary labels.

Drawing was unlabeled.

Quality of Equation

Showed every step of equation with no mistakes.

Skipped a step in equation but can be understood.

Able to understand and follow along with the equation but made 1 mistake.

Skipped 1 step of equation with 2-3 mistakes.

Did not have an equation or had 4+ mistakes.

Respectful Behavior/ Responsibility

Showed respectful behavior, kept team members on task and was a leader in the project.

Showed respectful behavior to group members, teacher and hallway rules. Participated in the project.

Participated in the group project but disrupted the group 1-3 times.

Participated half the time in the project but disrupted the group 3-6 times.

Did not participate in the project, and disrupted the research.

Real Life Application/ Detailed Explanation

Provided detailed paragraph of 2 real life applications of their right triangle.

Provided a detailed paragraph of 1 real life application of their right triangle.

Provided a paragraph (5 sentences) of 1 real life application of their triangle.

Provided 2-3 sentences of real life application of right triangle.

Provided a sentence of real life application of their right triangle.

Presentation Groups explained their project in detail and all members contributed.

Groups explained their project well with all members contributing.

Groups explained there project enough to understand and all members contributed.

Groups explained their project with 1-4 mistakes, all by one contributed.

Groups did not explain project well and 1-2 members contributed.

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7.2 Converse of the Pythagorean Theorem

Grade Level: 10th


Materials Needed: Whiteboard, markers, smart board, textbook, pencil, paper

Standards:



Objectives: Given any triangle, students will be able to identify if it is a right,

acute or obtuse triangle. Students will be able to state the difference between the

Pythagorean Theorem and the Converse.

Learning Activities:Review: Pythagorean TheoremConverse to the Pythagorean Theorem

What is another way we could use the Pythagorean theorem? What if I give you three positive numbers, and ask you if this is a right

triangle? Could you do that? Group Discussion This what the Converse to the Pythagorean Theorem is.

o Is used to verify that a given triangle is a right triangle.o If the square of the lengths of the longest side of a triangle is

equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Examples: Quizizz.com- Converse to the Pythagorean Theorem

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MathIsFun: https://www.mathsisfun.com/geometry/triangles-interactive.html

Tell whether the given triangle is a right triangleo 9 , 15, 3sqrt(34). Walk though how to set this up with the

students. Identify the longest side and shorter sides. YESo 14, 22, 24. Have students identify the longest side, then the

two shorter ones. Have students tell me how to set up the problem. NO

Can verify if a triangle is acute or obtuse.o Acute: If the square of the lengths of the longest side of a

triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle. c2<a2+b2 then it is an acute triangle.

o Obtuse: If the square of the lengths of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is a obtuse triangle. c2>a2+b2 then it is an obtuse triangle.

Examples: Classify if the triangle is right, acute or obtuse.o 4.3, 5.2, 6.1. Show students, identify longest side, set up

inequality, solve. Acute Triangle.o 6.5, 7.1, 11.2. Have students work this one out in notes, have

them walk me through this problem. Obtuse Triangle.

Optional: Have students do their own problem, where they make a right, acute or obtuse triangle. Have some or all students present them to the class. OR Write on the board RIGHT, ACUTE, or OBTUSE, then have students identify a triangle that fits that description. Have some or all students present to the class.

Homework: 1-34

Assessment: Formal Assessment: Quiz/Test

o Quiz taken in three dayso Test taken in eleven days or at end of chapter

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Informal Assessment: Have students individually or groups create a word problem that incorporates the use of the Converse to the Pythagorean Theorem.



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7.4 Special Right Triangles

Grade Level: 10th


Materials Needed: Whiteboard, markers, smartboard, textbook, paper, pencil

Standards:G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.


Objectives: Students will be able to identify the two special triangles and their

properties they include. Students will be able to solve a special triangle missing a side without

using the Pythagorean Theorem.

Learning Activities:Special Right Triangles

Do you know of any special right triangles? Since they are right, there has to be a 90 degree angle but can you think of a triangle that is special? Group Discussion : Rafters

o 45-45-90 and a 30-60-90. Why would these two triangles be “special?” Lets take a look

45-45-90 degree Triangle.o To construct a 45-45-90 degree triangle, take a square, then

draw in the diagonal, then you have a 45-45-90 degree triangle.

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o What is so special about a 45-45-90 degree triangle? The legs are the same lengths so, that has to have some kind of relationship on the hypotenuse.

o The hypotenuse is sqrt(2) times as long as each leg. This is true in any 45-45-90 degree triangle where the legs are the same length.

o Examples: Right Triangle Angle of 45 and leg 8. Do we know this a 45-45-90

degree triangle? How so? What is the measure of the other leg? Since this is 45-45-90 degree triangle, then the length of the hypotenuse is 8sqrt(2).

Both legs 3sqrt(2). How do we know that this is a 45-45-90 degree triangle? Both legs are the same lengths, so both of the angles must be equal, then it is a 45-45-90 degree triangle. How do we find the length of the hypotenuse? Take 3sqrt(2)*sqrt(2)= 6.

Hypoentuse 5sqrt(2), legs x. x=5o That is one of the special triangles, now what is the other one?

30-60-90 triangle 30-60-90 Triangle

o Contruct a 30-60-90 triangle by dividing an equilateral triangle in half. What are the properties of an equallateral triangle? What happens to the angle and side when we drop the perpendicular? This has to has some relation right?

o The hypotenuse is twice as long as the shorter leg, and the longer leg is sqrt(3) times as long as the short leg.

o So in the triangle we constructed, if the side lengths of the equilateral triangle is 6, what the length of the bottom side, and the height? Height= 3*sqrt(3)=5.2

o Examples: Right Triangle Triangle with 30-60-90 degrees and leg 9 and x,

hypotenuse y. longer leg is 9, shorter leg is x. x=3sqrt(3), y=2*3sqrt(3)=6sqrt(3)

30-60-90 triangle with shorter leg sqrt(3) and longer leg x. x=sqrt(3)*sqrt(3)=3

Equilateral triangle with sides 4. h=2sqrt(3)

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Quizizz: https://quizizz.com/admin/search/special%20right%20triangels?cloned=false

Review: What are the properties of a 45-45-90 triangle? hypotenuse is sqrt(2) times as long as each leg. 30-60-90 triangle? shorter leg x, longer leg xsqrt(3), hypotenuse 2x.

Need more help, come back to the back table.

Homework:1-33


o Quiz taken in two dayso Quiz taken in ten days

Informal Assessment: Interview by stating properties of special triangles, Board Work



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https://quizizz.com/admin/search/special%20right%20triangels?cloned=false

https://quizizz.com/admin/search/special%20right%20triangels?cloned=false

6.) 7.)

8.) 9.) 10.)

xx

x

x

x

x

55 14

2

25

2636

64°26°

42°

62°

39°72°

7.1-7.4 Quiz: Name:Read ALL the directions for each section, CIRCLE your answer.

Fill in the Blank. For questions 1- 4 refer to ∆ ABC , answer these statements with line segments such as AB.

1.) tan A=

2.) sin A=

3.) cos B=

4.) tan B=

Find the value of x to the nearest tenth.

5.)

13

A

B

C

22

37°

18

11

14

51°

4.5

26°

7

31.6°

20°

12

11.) 12.) 13.)

14.) 15.) 16.)

Q R

S

A

BC

X

Y

Z

L

J

K

M

NP

V

Q

P

Chapter 7: 7.1, 7.2, 7.4 Quiz Name:Solve the right triangles. Round decimal answers to the nearest tenth.

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17.) Describe the difference between an angle of depression and an angle of elevation.

17.) Drive in Movie: You are 50 feet from the screen at a drive-in movie. Your eye is on a horizontal line with the bottom of the screen and the angle of elevation to the top of the screen is 58 degrees. How tall is the screen? Draw a diagram.

18.) Airplane Landing: You are preparing to land as airplane. You are on a straight line approach path that forms a 3 degree angle with the runway. What is the distance s along this approach path to your touchdown point when you are 500 feet above the ground? Round your answer to the nearest foot.

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7.5 Apply the Tangent Ratio

Grade Level: 10th


Materials Needed: Paper, pencil, whiteboard, markers, smart board, Magazine Article

Standards:


G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems

Objectives: Apply the tangent ratio in any right triangle to find the missing side. Be able to recall the properties of tangent with no difficulty.

Learning Activities: Attention Getter: Leaning Tower of Pisa! How “tall” is the building?

Can solve this with Pythagorean theorem and Tangent. Can you find another structure that pertains to triangles?

Define Terms:o Trigonometric Ratio

Ratio of the lengths of two sides in a right triangle. What is a Ratio? Relationship between two numbers Use the trig. Ratios to find the measure of a side or an

acute angle.o Tangent

Ratio of the length of the legs in a right triangle Is constant for a given triangle

o What exactly is tangent in a triangle?

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o Θ (Theta) is a symbol that represents an angle measureo tan Θ= length of leg opposite of angle Θ / length of leg

adjacent to angle Θ tan Θ= opposite/adjacent remember Θ is any angle measure, but tangent involves

an acute angle.o Show tangent in a right triangle ABC for angles A and B.o Show the opposite and adjacent sides of the given angle.o What is tangent? Opposite over adjacento Check for clarity

Do some Examples of right triangles:o 18, 80, 82 for lengths of sides of the triangle.

Do each step fully, slowly, explain each thing twice, check for clarity.

Show students how to enter tan into their calculators or phones

o 8, 15, 17 for length of sides of the triangle. How do I do this problem for each angle? Have students walk you through how to do it.

o Find value of x- 32 degrees, opposite side 11, adjacent x. Do each step fully, slowly, explain each step twice, check

for clarity.o Find value of x- 65 degrees, opposite side x, adjacent side 24

Do this example with students.o Find value of x- 17 degree angle, opposite side 9, adjacent side

x Have students do this problem in their notes, then ask

someone to walk me through this problem or come up to do it on the board.

Assessment:o Objective 1: Formal Assessment: Test or Quiz

o Quiz in three dayso Test in seven days

o Objective 2: Informal Assessment: Interview, board work, discussionReflection:

What did I do well?

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If I were to teach this lesson again, what would I keep the same? If I were to teach this lesson again, what would I change? This is completed after your lesson is taught.


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7.6 Apply the Sine and Cosine Ratio

Grade Level: 10th


Materials Needed: Paper, pencil, whiteboard, markers, smart board

Standards:


G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.


Objectives: Apply the sine and cosine ratios in any right triangle to find the

missing side or angle. Be able to recall the properties of sine and cosine.

Learning Activities: Recall Tangent

o What was the ratio? tan A= opposite/ adjacent

Sine and Cosine Ratioso Similar to tangent ratio but not the sameo Trigonometric ratios for acute angles that involve the lengths

of a leg and the hypotenuse of a right triangleDefine sine and cosine:Given a right triangle ABC; use AB, AC, and BC to define the following:

sineo sin A= opposite/ hypotenuse

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o Does everyone understand? cosine

o cos A= adjacent/hypotenuseo Does everyone understand?

sin B and cos B Lets do some Examples: Examples: Build up

o Hypotenuse=65, leg-16, leg-63. Find sin R and sin S; then cos R and cos S. (I do)

o Hypotenuse=25, leg=15, leg=20 Find sin X and sin Y; then find cos X and cos Y. (Students tell me what to do)

o Hypotensue= ? , leg= 9, 12. Find cos R and cos S; then sin R and sin S. (different students tell me what to do)

o Hypotenuse= 55, angle= 64. Find legs x and y. (student comes up to smartboard)

o Hypotenuse= 14, angle= 26. Find legs x and y. (if necessary, different student comes up to board)

Why can’t I do the tan, sin, or cos of angle C?o We already know that angle being a right angle.

What is sine? What is cosine? What is tangent?o Multiple times to students, different students answer

Exit slip: What is sine? What is cosine? What is tangent? What do they have in common? How are they different?

Homework: 1-27, 30, 31, 33, 34

Assessment:o Objective 1:Formal Assessment: Test or Quiz

o Quiz in two dayso Test in six days

o Objective 2: Informal Assessment: Call on student to say the trigonometric ratios.


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7.7 Solving Right Triangles

Grade Level: 10th


Materials Needed: Whiteboard, smart board, markers, pencil, paper

Standards:


Objectives: Students will be able to solve right triangles using either the

Pythagorean Theorem or trigonometric ratios. Students can find all the angles and sides measure of a right triangle

given two sides; or a side and a acute angle measure.

Learning Activities:Magazine/article Example: Solve Right triangle

Review Pythagorean Theorem, Tangent, Sine and Cosine. To solve a right triangle means to find the measure of all of its sides

and angles, you need either:o Two sideso One side length and the measure of one acute angle.

Inverse trigonometric Ratioso Just like the sound, the inverse of the trigonometric ratios.o Inverse tangent:

If tanA=x, then tan-1x = m angle Ao Inverse Sine

If sinA=y, then sin-1y = m angle Ao Inverse Cosine

If cosA=z, then cos-1z = m angle A

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o Define them by drawing a triangle on the board. Examples:

o Right triangle with legs 15, 20. tanA= 15/20 so tan-1(15/20)=Ao Given sin A= .87 then A= sin-1(.87)= 60.5 degreeso Given cos B= .15 then B= cos-1(.15)=81.4o A stage that is a right angle from the side with height 2 and

hypotenuse 30. How long is the stage and what are the acute angles?

Have students walk me through this. What do I find first? Does it matter? Smallest angle is 3.824.

Exit slip: What is the difference between the trigonometric ratios and their inverses? How does this help you solve right triangles?

Homework: 1-34.


o Quiz in one dayo Test in five days

Informal Assessment: Project tomorrow



23

6.) 7.)

8.) 9.) 10.)

xx

x

x

x

x

55 14

2

25

2636

64°26°

42°

62°

39°72°

Chapter 7: 7.5-7.7 Quiz Name:Read ALL the directions for each section, CIRCLE your answer.

Fill in the Blank. For questions 1- 4 refer to ∆ ABC , answer these statements with line segments such as AB.

1.) tan A=

2.) sin A=

3.) cos B=

4.) tan B=

Find the value of x to the nearest tenth.

5.)

24

A

B

C

22

37°

18

11

14

51°

4.5

26°

7

31.6°

20°

12

11.) 12.) 13.)

14.) 15.) 16.)

Q R

S

A

BC

X

Y

Z

L

J

K

M

NP

V

Q

P

Solve the right triangles. Round decimal answers to the nearest tenth.

25


17.) Drive in Movie: You are 50 feet from the screen at a drive-in movie. Your eye is on a horizontal line with the bottom of the screen and the angle of elevation to the top of the screen is 58 degrees. How tall is the screen? Draw a diagram.

18.) Airplane Landing: You are preparing to land as airplane. You are on a straight line approach path that forms a 3 degree angle with the runway. What is the distance s along this approach path to your touchdown point when you are 500 feet above the ground? Round your answer to the nearest foot.

26

Article ProjectStudents will research an article that contains the concepts we are talking about in this chapter: Pythagorean Theorem, Tangent, Sine and Cosine. Student will present their article to the class and also write a mini paper about their finding and their thoughts.

5 4 3 2 1Article Choice Selected a

detailed article with 2 concepts from the chapter with in depth real world application.

Selected an article with a concept from the chapter with in depth real world application.

Selected an article with a concept from the chapter that is in depth.

Selected an article that mentions a concept for the chapter but not in depth.

Selected article that did not pertain to the concepts from the chapter.

Presentation Material

Presented all material in the article and explained the concepts in depth.

Presented most of material in article with in depth explanation.

Presented relevant material in article with enough detail to understand.

Presented concept material in the article with explanation but did not understand concept.

Presented irrelevant material in the article with no explanation of the concept.

Presentation Easy to understand and confident about content in article.

Able to follow understanding with effort and comfortable with content in article.

Able to understand and comfortable with content in article.

Hard to understand, but could be understood, and wiry about content in article.

Unable to understand and uneasy about content in article.

Mini Paper Included and explained content and concept in article, reflected on findings.

Included content and concept in article with little explanation and reflected on finding.

Included content and concept in article.

Included content in article but did not include concept.

Did not explain content and concept in article, no reflection.

27

SOH-CAH-TOA Checklist

Create either SOH-CAH-TOA story, song, or word problem where you solve using sine, cosine or tangent.

Stayed on task

Worked with group members

Presented the material with group

Material involved SOH-CAH-TOA

Respectful

28

x

x

y

Chapter 7 Test Name: Show ALL your work for each problem. CIRCLE your answer.Find the value of x. Write answers in simplest radical form.

Is this a right triangle? If so label right, acute or obtuse.

5.) 5,7,9 6.) 3,5, √34 7.) 6, 8, 16

8.) 3.1 , 4.5 , 5.2 9.) 9, 15, 10√3 10.) 3, 4, 5

Find the value of each variable. Write your answers in simplest radical form.

29

11.)

x

20

151.) 2.) x

10

4

3.) 4.)x

38

18

12

3y

2x-86√2

13.)12.)

20

32°

22.)

B

AC

6

8

23.)E F

G

14.5

61.4°

Find the value of x. Round your answer to the nearest tenth.

Solve the right triangle; find all missing side lengths and angle measures. Round your answer to the nearest tenth.

30

14.) 15.) 16.)

17.) 18.) 19.)

x x x

xx

x

5

40°

12

65° 18

31°

55°

15

18

31°


23.) A soccer ball is placed 10 feet away from the goal, which is 8 feet high. You kick the ball and it hits the crossbar along the top of the goal. What is the angle of elevation of your kick? Round to nearest tenth.

24.) Describe how you would decide whether to use the Pythagorean Theorem or trigonometric ratio to find the unknown sides of a right triangle.

25.) You are cleaning the gutters on your house. Some bushes extend 7 feet from the wall of the house. The gutters are at a height of 24 feet. What is the minimum length of ladder you will need? If you have a 50 foot ladder, what is the minimum angle the ladder can form with the ground? Round to nearest hundredths.

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summerbailey.weebly.com€¦ · Web vieweither SOH-CAH-TOA story, song, or word problem where you solve using sine, cosine or tangent: Groups . Day 12: Review /Reflect. for Chapter

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