1 UNIT 8 RIGHT TRIANGLES NAME PER I can define, identify and illustrate the following terms leg of a right triangle short leg long leg radical square root hypotenuse Pythagorean theorem Special Right Triangles Trigonometry Reference Angle Adjacent Opposite Sine Cosine Tangent Holiday Holiday 8 Pythagorean Theorem 15 Mixed practice 22 Trigonometry 9-10 Pythagorean Theorem Trigonometry REVIEW Begin Test 11 Isosceles Right Triangles 18 Trigonometry 25 TEST Tuesday, 1/8 Pythagorean Theorem 1. I can solve for the missing hypotenuse of a right triangle. 2. I can solve for the missing leg of a right triangle. 3. I can identify Pythagorean Triples. ASSIGNMENT: Introduction to Pythagorean Theorem Worksheet Grade: Block day, 1/9-10 Pythagorean Theorem, Converse, and Inequalities 4. I can use the Converse of the Pythagorean Theorem to determine if a triangle is a right triangle or not. 5. I can determine if a triangle is acute or obtuse using the Pythagorean Inequalities theorem. ASSIGNMENT: Pythagorean Theorem Converse and Inequalities Worksheet Grade: Friday, 1/11 Isosceles Right Triangles ( ) I can solve for the 2 missing sides of an isosceles right triangle. ASSIGNMENT: Isosceles Right Triangle Worksheet Monday, 1/ Triangles I can solve for the 2 missing sides of a ASSIGNMENT: Worksheet Tuesday, 1/15 Mixed Practice I can choose the correct method to solve a right triangle problem. I can solve problems using Pythagorean Theorem and/or Special Right Triangles. ASSIGNMENT: Mixed Practice Worksheet Grade: Grade: Grade:
2 Block day, 1/16-17 Trigonometry I can write the trigonometric ratios. I can solve problems using trigonometric equations. I know the relationships between sine, cosine, and tangent. ASSIGNMENT: Introduction to Trig Worksheet Friday, 1/18 Trigonometry I can write the trigonometric ratios. I can solve problems using trigonometric equations. I know the relationships between sine, cosine, and tangent. ASSIGNMENT: Introduction to Trig Worksheet Monday, 1/22 Trigonometry I can find another trig function, given one. I can find multiple pieces of a triangle using trigonometry. ASSIGNMENT: More Trig Worksheet Grade: Grade: Grade: Block day, 1/23-24 I can do all above objectives. ASSIGNMENT: Review Worksheet Review Grade: Friday, 1/25 TEST #8: Right Triangles Test #8 I can demonstrate knowledge of ALL previously learned material. TEST #8: Right Triangles Grade:
3 Notes: Introduction to Pythagorean Theorem Previous Knowledge: 1) The largest side of a triangle is across (opposite) from the. 2) The of a right triangle is always across from the. 3) The Pythagorean Theorem is. And c is always used for the. Ex. 1) What variable represents the hypotenuse? w r p b) If p = 8 and r = 15 then w =. Ex. 2) What variable represents p b) If p = 25 and r = 24 then w =. the hypotenuse? w r 3. A 4. T mi 5. k 9 ft. N r R 41 ft. I 2 mi A M x 10 yd. T 18 yd. A A) 424 B) C) D) Find the missing side x y 6 3 5
4 NAME DATE PER. Introduction to Pythagorean Theorem Assignment Use the Pythagorean Theorem to find the missing length. Give answers to nearest hundredth. 1. a = 8 and b = a = 24 and c = 28. Solve each problem. Round to the nearest hundredths x x x x 5 2 x The slide at the playground is 12 feet tall. If the d bottom of the slide is 15 feet from the base of the 5 ladder, how long is the slide? 6 3 Page 1 of 2 (continue on)
5 9. If you place a 16 ft ladder 6 feet from 10. A tree broke 6 feet from the bottom. a wall, how high up the wall will it go? If the top landed 12 feet from the base, how tall was the tree before it broke? 11. Jim headed south 5 miles from his house 12.There is a restaurant diagonally across to the cleaners. From there he headed west a rectangular field from Jeff s dorm. If to meet his friends. They were at a park 3 he followed the roads, he would have to miles away. How far would he have to go go 2 blocks north and 3 blocks east. if he went straight home? Each block is 100 ft long. How much shorter would it be for him if he walked diagonally across the field instead? MULTIPLE CHOICE: Find the correct answer for each of the following. Clearly circle your answers. WORK MUST BE SHOWN IN ORDER TO RECEIVE CREDIT. 13. If KMP is a right triangle formed by A 159 in. 2 B 129 in. 2 C 66 in. 2 D. 24 in The figure below shows three right triangles joined at their right-angle vertices to form a triangular pyramid. Which of the following is the closest to the length of XZ? A. 7 inches B. 20 inches C. 12 inches D. 9 inches 13 in X Y W 12.5 in 15 in Z 15. The legs of a right triangle are 4 cm and 7 cm long. To the nearest cm, how long is the hypotenuse? A. 11 cm B. 10 cm C. 14 cm D. 8 cm 16. What is the height of the triangle? 3 cm h 2 cm 5 cm A. 2 cm B. 1 cm C. 5 2 cm D cm Page 2 of 2 (STOP)
6 Notes: Pythagorean Theorem Converse and Inequalities The Pythagorean Theorem states: If a triangle is a right triangle, then the sum of the squares of the lengths of the two legs of the triangle is equal to the square of the hypotenuse. Write the converse:. In your own words what does the converse let you do? Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. A) 8, 11, 13 B) 5,7, 10 C) 5,8, 17 APPLICATION PROBLEMS Ex1 What is the Question? What do you need to know? What other information do you need to know or what do you need to use? How do you solve the problem?
7 Ex 2 What is the Question? What do you need to know? What other information do you need to know or what do you need to use? How do you solve the problem? Ex 3 What is the Question? What do you need to know? What other information do you need to know or what do you need to use? How do you solve the problem? Ex 4 What is the Question? What other information do you need to know or what do you need to use? What do you need to know? How do you solve the problem? A yield sign is in the shape of an equilateral triangle. Each side is 36 inches. Which of the following measurements best represents the area of the yield sign?
8 NAME DATE PER. Pythagorean Theorem Converse and Inequalities Assignment Determine if a triangle can be formed with the given lengths. If so, classify the triangle by angles. 1. 7, 20, and 12 YES or NO Classify: 2. 15, 8, and 17 YES or NO Classify: 3. 12, 10, and 8 YES or NO Classify: 4. 20, 8, and 19 YES or NO Classify: 5. 16, 30, 34 YES or NO Classify: 6. 80, 71, and 5 YES or NO Classify: Find the indicated length A rectangle has a diagonal of 2 and 3 a length of 3. Find its width. 6 x 9. Find the length of a diagonal of a 10. If you had a 20 ft ladder, how far away square with perimeter 16. from a building would you have to place the bottom to reach a window 15 feet up? Page 1 of 2 (continue on)
9 11. Jenny has a rectangular shaped 12. A twelve foot pole broke back yard. It is 15 feet wide and 23 7 feet from the top. How far feet diagonally. If she wants to plant trees from the base of the pole 2 feet apart all the way around her yard, did the top land? how many trees would she need? MULTIPLE CHOICE: Find the correct answer for each of the following. Clearly circle your answers. Work must be shown in order to receive credit. 13. Which of the following sets of numbers is a Pythagorean triple? A B C D. Not Here 16. Use the Pythagorean theorem to find the figure that is a right triangle. 14. If 24 and 40 are two of the three numbers of a Pythagorean triple, what is the third number? A A. 48 B. 16 C. 32 B D Refer to the right triangle shown below. Find the value of x. C D. A B C D. 2 Page 2 of 2 (STOP)
10 NOTES: Isosceles Right Triangles A diagonal of a square divides it into two congruent. Since the base angles of an isosceles triangle are, the measure of each acute angle is. So another name for an isosceles right triangle is a triangle. A triangle is one type of. FROM STAAR CHART Example 1A: Finding Side Lengths in a 45-45º- 90º Triangle Find the value of x. Example 1B: Find the value of x. Example 2: Jana is cutting a square of material for a tablecloth. The table s diagonal is 36 inches. She wants the diagonal of the tablecloth to be an extra 10 inches so it will hang over the edges of the table. What size square should Jana cut to make the tablecloth? Round to the nearest inch.
11 Name: Period: Isosceles Right Triangles Assignment I. Fill in the length of each segment in the following figures t y 45 2x 6 2x 5 Page 1 of 2 (continue on)
12 16. Sam has a square backyard divided into 2 sections along the 40 foot diagonal. One of these sections is used as a garden. What is the approximate area of the garden? 17. A guy wire supporting a radio tower is positioned 145 feet upthe tower. It forms a 45 anglewith the ground. About how longis the wire? Find the perimeter and area of a triangle with a hypotenuse length 12 inches. 20. This triangle loom is made from wood strips shaped into a triangle. Pegs are placed every inch along each leg. Suppose you make a loom with an 18-inch hypotenuse. Approximately how many pegs will you need? Find the value of x in simplest radical form. 24. Given AC = 10, find BX in simplest radical form. B A X C Page 2 of 2 (STOP)
13 Notes: FROM STAAR CHART A triangle is another. You can use an triangle to find a relationship between the lengths. Example 1A: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form. Example 1B: Find the values of x and y. Give your answers in simplest radical form. Example 1C: Find the values of x and y. Give your answers in simplest radical form. Example 1D: Find the values of x and y. Give your answers in simplest radical form.
14 Name: Period: Triangles Assignment Fill in the blanks for the special right triangles t y 0 7. RJQ is equilateral. 8. ABC is equilateral. J B JQ = AD = 4 3 RL = h DC = LQ = AB = R6 L Q JL = A D C BC = Page 1 of 2 (continue on)
15 21. The hypotenuse of a triangle is 12 2 ft. Find the area of the triangle. 22. Find the perimeter and area of a triangle with hypotenuse length 28 centimeters. 24. Find the perimeter and area of an equilateral triangle with height 30 yards. 25. A skate board ramp must be set up to rise from the ground at 30. If the height from the ground to the platform is 8 feet, how far away from the platform must the ramp be set? 8 ft Find the value of x. Page 2 of 2 (STOP)
16 Name: Period: Mixed Practice Assignment I. For each problem: 1) Determine if you should use Pythagorean Theorem, , or ) Show work and find all the missing segment lengths O 1. Use: 2. Use: C W 3. Use: 4. Use: ABC is equilateral with perimeter 36y units. Find the length of each side and the height. A Use: B D C 6. **C is the center of a regular hexagon. Find the length of each side. Use: 30 C Hexagons are made of 6 equilateral triangles. 71 When viewed from above, the base of a water fountain has the shape of a hexagon composed of a square and 2 congruent isosceles right triangles, as represented in the diagram below. Which of the following measurements best represents the perimeter of the water fountain s base in feet? A ( ) ft C ( ) ft B ( ) ft D ( ) ft Page 1 of 3 (continue on)
17 Alex has a square garden in his back yard. If the 11. FGH is an equilateral triangle. garden has a diagonal of 18 inches, what is the area Which value is closest to the perimeter of FGJ? of Alex s square garden? A 39 in B 52 in C 62 in D 66 in 12. Nicole is creating a support in the shape of a right triangle. She has a 92 cm-long piece of wood, which is to be used for the hypotenuse. The two legs of the triangular support are of equal length. Approximately how many more centimeters of wood does Nicole need to complete the support? A 130 cm B 184 cm C 260 cm D 276 cm 13. Two identical rectangular doors have glass panes in the top half and each bottom half is made of solid wood. If 1 meter is approximately equal to 3.28 feet, what is the approximate length of x in feet? A 5.3 feet B 8.5 feet C 2.6 feet D 7.1 feet 3.2 m 4.1 m Page 2 of 3 (continue on)
18 14. Which of the following could be the side lengths of triangle? A 2 in, 4 in, 2 2in B 2 in, 4 in, 2 3in C 2 in, 2 in, 2 2in D 4 in, 4 in, 4 3 in A cube with side lengths of 4 inches is shown below. How could you find the length of d, the diagonal of the cube? A (4 3) = d B (4 3) = d C D = d = d 19. Jenna is flying a kite on a very windy day. The kite string makes a 60 angle with the ground. The kite is directly above the sandbox, which is 28 feet away from where Jenna is standing. Approximately how much of the kite string is currently being used? A 56 feet B 48.5 feet C 40 feet D 14 feet Page 3 of 3 (STOP)
19 Notes Introduction to Trig The mathematics field called Trigonometry is the study of triangles and the ratios of the sides. Each angle of a right triangle has a unique decimal value for each trigonometric ratio. Your calculator has these tables memorized for you. Find the SINE, COSINE and TANGENT buttons on your calculator. 1) Press and make sure the selection is highlighted. Always check that your calculator is in DEGREE mode. You are responsible to check. 2) Press the Trigonometric function you would like followed by the measure of the angle. Round to the nearest hundredth. Ex 1. sin 35 = Ex 2. cos 18 = Ex 3. tan 87 = If you are given the ratio and asked for the angle, you just use the ratio backwards. Your calculator needs to be told to do this. Write the keys you will press and then write the angle to the nearest degree. Ex 7. 8 sin x = x = Ex 8. tan x = x = Ex cos x = x = 2 There are 3 of trigonometric relationships that we study. Sine is the ratio of the side to the. Cosine is the ratio of the side to the. Tangent is the ratio of the side to the side. The NEVER changes, but and are dependent on the used. The angle is NEVER used. The three sides of the triangles are referred to as Hypotenuse (H), Adjacent (A), and Opposite (O). Label each side of each triangle using angle W as your reference. Ex 1. Ex 2. Y Z Ex 3. W W W Z Y
20 To help you remember these relationships, you can use the phrase. The trigonometric ratios are written in an equation form. (**Hint: Write these ratios at the top of EVERY page you are working on.) Sine x = Cosine x = Τ angent x = USE THE TRIANGLE AT THE RIGHT to determine the following trigonometric ratios. Ex 4. sin 40 = Ex 5. sin a = a Ex 6. cos 40 = Ex 7. cos a = 10 n Ex 8. tan 40 = Ex 9. tan a = 30 40º Use the triangle at the right to write all of the following trigonometric equations. From 72 From x 55 18º Use Trigonometric Ratios to Solve for Missing Sides and Angles 1) Determine which Trig Ratio will fit your information. 2) Set up the Trig Ratio 3) Round to the nearest degree if it is an angle and round to the nearest hundredth for sides. Ex n Ex2. Ex z w
21 Ex 4. Ex x k Ex 5. w Angle of depression - Angle if elevation - Ex 1. Angie looks up at 25 degrees to see an airplane flying toward her. If the plane is flying at an altitude of 3.5 miles, how far is it from being directly above Angie? Picture: Equation: Solution: Ex 2. A six foot vertical pole casts a shadow of 11 feet. What is the angle of elevation with the ground? Picture: Equation: Solution: Ex 3. Lauren is at the top of a 15 m lookout tower. From an angle of depression of 25, she sees Evan coming toward her. How far is Evan from the base of the tower? Picture: Equation: Solution:
22 Name Period Introduction to Trig Worksheet Write down what buttons you would push to get the answer for the following problems. Do not forget to include checking the mode. Then answer the problems. 1. sin 40 = 2. 5 tan x = x = 7.5 The three sides of the triangles are referred to as Hypotenuse (H), Adjacent (A), and Opposite (O). Label each side of each triangle using angle D as your reference. D F D F E D E Use the triangles below to write all 6 trig equation from the two acute angles º 5. aº 10 n m 40º 48 bº Page 1 of 3 (continue on)
23 Set up and Solve for variable round angles to the nearest degree and sides to the nearest hundredths. 6. x = 7. n = x n 8. z = 9. w = z 18 6 w 10. z = 11. x = x z 12. w = 13. k = 8.7 w k d = 15. w = 12 d w Page 2 of 3 (continue on)
24 1. A kite is flying at an angle of 63 with the ground. If all 250 feet of string are out, and there is no sag in the string, how high is the kite? 2. A 30 foot tree broke from its base and fell against a house. If the tree touches the house at 21 feet, what angle is the tree forming with the house? 3. A tree casts a shadow of 28 m. The elevation of the sun is 49. How tall is the tree? 4. Joey is putting up an antenna. At the 30 foot mark, he attaches a 50 foot guy wire. What angle does the guy wire form with the antenna? 5. A freeway entrance ramp has an elevation of 15. If the vertical lift is 22 feet, what is the distance up the ramp? 6. A person at the top of a cliff 100 feet tall sees Gilligan s boat. His sighting of the boat is at an angle of depression of 10. How far is the boat from the base of the cliff? 7. A 24 foot ladder is placed against a wall at 55 with the ground. How far away from the wall is the base of the ladder? 8. A 32 in. bat is leaning against a fence. If the bat is 15 in. away from the base of the fence, what angle is formed between the ground and the bat? 9. Ana knows that she is one mile from the base of a tower. Using a protractor she estimates an angle of elevation to be 3. How tall is the tower to the nearest foot? (1 mile = 5280 feet) 10. A plane takes off at an elevation of 20. In its path is the tower of 170 feet. If the plane at takeoff is 500 feet away from the tower. What is the altitude of plane? Will it clear the height of the tower? If yes, by how much? Page 3 of 3 (STOP)
25 Name Period 1. Trig Worksheet Page 1 of 3 (continue on)
26 7. A surveyor 50 meters from the base of a cliff measures the angle of elevation to the top of the cliff as 72. What is the height of the cliff? Round to the nearest meter. 8. Grand Canyon Problem: From a point on the North Rim of the Grand Canyon, a surveyor measures an angle of depression of 1º to a point on the South Rim. From an aerial photograph, he determines that the horizontal distance between the two points is 10 miles. How many feet is the South Rim below the North Rim to the nearest foot? (Note: 1 mile = 5280 feet) 9. At a point 125 feet from the base of a building, the angle of elevation to the third floor is 22 and to the 10 th floor is 53. How much higher is the tenth floor than the third floor? 12. Page 2 of 3 (continue on)
27 Page 3 of 3 (STOP)
28 Name Period Review Assignment Page 1 of 3 (continue on)
Class: Date: Right Triangles Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the length of the missing side. The triangle is not drawn
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Trigonometry Module T09 Solutions of Right Triangles with Practical pplications Copyright This publication The Northern lberta Institute of Technology 2002. ll Rights Reserved. LST REVISED December, 2008
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Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
Geometry CC Assignment #1 Naming Angles 1. Name the given angle in 3 ways. A. Complementary angles are angles that add up to degrees. 3. Supplementary angles are angles that add up to degrees. #1 4. Given
2.1 The Tangent Ratio In this Unit, we will study Right Angled Triangles. Right angled triangles are triangles which contain a right angle which measures 90 (the little box in the corner means that angle
1. A person has 78 feet of fencing to make a rectangular garden. What dimensions will use all the fencing with the greatest area? (a) 20 ft x 19 ft (b) 21 ft x 18 ft (c) 22 ft x 17 ft 2. Which conditional
Find the exact values of the six trigonometric functions of θ. 1. The length of the side opposite θ is 8 is 18., the length of the side adjacent to θ is 14, and the length of the hypotenuse 3. The length
EXERCISES: Triangles 1 1. The perimeter of an equilateral triangle is units. How many units are in the length 27 of one side? (Mathcounts Competitions) 2. In the figure shown, AC = 4, CE = 5, DE = 3, and
Perfume Packaging Gina would like to package her newest fragrance, Persuasive, in an eyecatching yet cost-efficient box. The Persuasive perfume bottle is in the shape of a regular hexagonal prism 10 centimeters
Right Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle Preliminar Information: is an acronm to represent the following three trigonometric ratios or formulas: opposite
Find x. 1. of the hypotenuse. The length of the hypotenuse is 13 and the lengths of the legs are 5 and x. 2. of the hypotenuse. The length of the hypotenuse is x and the lengths of the legs are 8 and 12.
mathematics College Algebra Geometry Trigonometry Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about
Name: Period GL UNIT 11: SOLIDS I can define, identify and illustrate the following terms: Face Isometric View Net Edge Polyhedron Volume Vertex Cylinder Hemisphere Cone Cross section Height Pyramid Prism
BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that
CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY Specific Expectations Addressed in the Chapter Explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that
Santa Monica College COMPASS Geometry Sample Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the area of the shaded region. 1) 5 yd 6 yd
Find the volume of each prism. 3. the oblique rectangular prism shown at the right 1. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. If two solids have
Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary. 2. 1. Use the Pythagorean Theorem to find the height h, of the parallelogram. Each pair of opposite
Chapter 8 Further Applications of Trigonometry 1021 8.1 Non-right Triangles: Law of Sines In this section, you will: Learning Objectives 8.1.1 Use the Law of Sines to solve oblique triangles. 8.1.2 Find
Students: 1. Students understand and compute volumes and areas of simple objects. *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Review