Assignment. Pg. 567 #16-33, even pg 577 # 1-17 odd, 32-37

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1 Assignment Intro to Ch Da 1 8. Da 8. Da 1 8. Da Review Quiz 8. Da 1 8. Da 8. Etra Practice In-class project 8.6 Da Da Ch. 8 review Worksheet Worksheet Worksheet Worksheet Worksheet Worksheet Review Worksheet Worksheet Pg. 567 #16-, 6- even Worksheet pg 577 # 1-17 odd, -7 Worksheet (Be read for a quiz tomorrow) Worksheet Worksheet Review Worksheet Geometr Chapter 8 Learning Targets! B the end of the chapter, ou should be able to: Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to its hpotenuse Use the Pthagorean Theorem and its converse Use the properties of triangles and triangles Find trigonometric ratios using right triangles, and use these ratios to find angle measures in right triangles Identif and use Angles of Depression and Angles of Elevation to solve problems and find missing values Use the Law of Sines and Law of Cosines to find missing values in triangles 1

2 Ch. 8 Introduction L.T.#1: Be able to write radicals in simplest radical form! L.T.#: Be able to solve equations involving radicals! Quick Vocab: What is a radical? What is simplest radical form? Now, let s fold some socks! You can multipl numbers that are under radicals!

3 You can divide numbers that are under radicals! But, NEVER leave a radical sign in our denominator! You must RATIONALIZE it!! Now, let s solve some equations using radicals! Leave our answers in simplest radical form! + = 0 + = 10 = = 0

4 Did we meet the target? L.T.#1: Be able to write radicals in simplest radical form! L.T.#: Be able to solve equations involving radicals! Solve this equation and write our answer in simplest radical form! = 6 Section 8.1 ~ Geometric Mean! L.T.: Be able to find the geometric mean and use it to find unknowns in similar right triangles! Geometric Mean: when the values on one diagonal of a proportion are equal to each other E. 1: Identif the geometric mean in each of the following proportions. GM: = 5 = GM: GM: = 7 E. : Find the value of the geometric mean. = 16 5 =

5 E. : Find the geometric mean of: a. and 1 b. and 8 c. 15 and 0 The Geometric Mean in Right s! Theorem: In a right triangle, when ou draw an altitude to the hpotenuse, ou create three similar triangles! B C ACD ~ ADB ~ DCB A D The legs and altitude of a right triangle are the geometric means between the segments of the hpotenuse that. a c b z 5

6 Let s practice! E. : Refer to the picture to complete each proportion. = z a = c Challenge: b = a z = z b b b = a z b z = z c E. 5: Find the values of the variables

7 Just a few more! z Section 8. ~ da 1 The Pthagorean Theorem!! L.T.: Be able to find unknowns in triangles using the Pthagorean Theorem! Quick Warm-up: + = 15 + = 6 What do ou call the longest side of a right triangle? (the side opposite the right angle) 7

8 Pthagorean Theorem: In a triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hpotenuse. a b c a + b = Find the value of. Leave our answer in simplest radical form c Pthagorean Triples! A Pthagorean Triple is a set of whole numbers that satisfies the equation a + b = c. Common triples: Find the value of. 1 1 (Multiples of these sets are also triples!)

9 Did we meet the target? L.T.: Be able to find unknowns in triangles using the Pthagorean Theorem! Find the length of the hpotenuse! 9 1 Section 8. ~ da The Pthagorean Theorem!! L.T.#1: Be able to find unknowns in triangles using the Pthagorean Theorem, and find the areas of triangles! Quick Review: Find the value of each variable! Leave answers in simplest radical form What is an eample of a Pthagorean Triple? 9

10 Using Pthagorean Theorem to find AREAS! Find the area of the triangle. Do the legs form a triple? m 5 m A right triangle has a hpotenuse of length 5 and a leg of length 10. Find the area of the triangle in simplest radical form and also as a decimal. If a + b < c, then the is obtuse. If a + b > c, then the is acute. The lengths of the sides of a triangle are given. Is the triangle right, obtuse, or acute? Eplain. 1, 16, 0 1,, 1 11, 1, 15 11, 7, A window-washer leans a 1-foot ladder against the side of a building. The base of the ladder is 9 feet from the base of the building. How high up the side of the building does the ladder reach? 10

11 Did we meet the target? L.T.#1: Be able to find unknowns in triangles using the Pthagorean Theorem, and find the areas of triangles! Find the area of the triangle. m 6m Section 8. ~ da 1 Special Right Triangles! L.T.: Be able to find sides of triangles! Quick Review: Rationalize the following!

12 Find the length of the hpotenuse: Triangle Theorem: In a triangle, both are congruent (isosceles), and the length of the hpotenuse is times the length of each leg. Find the value of each variable:

13 Find the value of each variable! Find the area of the triangle!

14 Wh do we need to know about triangles? The are in the real world! Not to mention, it s a whole lot easier than using the Pthagorean Theorem. Yadier Molina wants to know how far he has to throw the ball to catch a man stealing second. ft 90 ft 90 ft Before, we had to use the Pthagorean Theorem. But now that ou know about triangles, ou can use the shortcut! Did we meet the target? L.T.: Be able to find sides of triangles! Find the value of

15 Section 8. ~ da Special Right Triangles! L.T.: Be able to find sides of triangles! Quick Review: Question: In a triangle, will an sides be the same length? Triangle Theorem: In a triangle, short the length of the hpotenuse is times the length of the shorter leg (short) the length of the longer leg (long) is times the length of the shorter leg (short) 60 hpotenuse 0 long No! 15

16 Find the value of each variable!

17 Let s practice some more! Find the area of each triangle A window-washer leans a 0-foot ladder against the side of a building. The base of the ladder make a 60 angle with the ground. How high up the side of the building does the ladder reach? 60 0 ft Find the value of each variable. d 60 7 b 5 c a 17

18 Did we meet the target? L.T.: Be able to find sides of triangles! Find the value of each variable! 5 60 a d 5 b c Section 8. da ~ Sine and Cosine Ratios! L.T.: Be able to use the sine and cosine ratios to find lengths and angles in triangles! The Sine Ratio: In a right triangle, the ratio of the leg OPPOSITE an angle to the is a constant. This is called the sine ratio! opp sin = hp C A B The Cosine Ratio: In a right triangle, the ratio of the leg ADJACENT to an angle to the is a constant. This is called the cosine ratio! cos = adj hp C A B 18

19 Let s practice! E. 1: Write the sine and cosine ratios for T. T V U T 1 U V E. : Find each of the following using our calculator! Round decimals to the nearest thousandth. sin 10 = cos 5 = sin15 = You can find an angle from a given sine or cosine ratio! These are called inverse sine and inverse cosine and ou can use the sin -1 and cos -1 buttons on our calculator! E. : Find the angle with the given ratio using our calculator! Round decimals to the nearest tenth. cos = sin = tan = E. : Use the triangle to find each of the following. a. sin A = b. cos A = A 5 1 c. sin B = d. cos B = C 1 B 19

20 Let s practice with the calculators! cos = sin 1 = tan =.751 cos8 = sin = 5 cos = Let s see if we ve met the learning target! E. 5: Use the appropriate ratio to find the value of each variable

21 Section 8. da ~ The Tangent Ratio! L.T.: Be able to use the tangent ratio to find lengths and angles in triangles! Quick Review: How do ou find the third side of a right triangle when ou alread know two sides? What are the two tpes of special right triangles we have worked with? What is trigonometr? Let s look at a right triangle! What side is the hpotenuse? A What is the leg opposite B? opposite What is the leg adjacent to B? C hpotenuse adjacent B The Tangent Ratio: In a right triangle, the ratio of the leg an angle to the leg to the same angle is a constant. This is called the tangent ratio! opp tan = adj C A B 1

22 Let s practice! E. 1: Write the tangent ratios for T and U. T 5 V 1 1 U E. : Find the tangent RATIO of each angle using our calculator! Round decimals to the nearest thousandth. 7 T 5 tan 10 = tan 5 = tan 0 = U V You can find an angle from a given tangent ratio! This is called an inverse tangent and ou can use the tan -1 button on our calculator! E. : Find the ANGLE with the given tangent ratio using our calculator! Round decimals to the nearest tenth. tan A =.1 tan B = 1. 8 E. : Use the triangle to find each of the following. a. tan A = A b. m A = c. m B = 6 C 15 B

23 Let s practice with the calculators! Round decimals to the nearest tenth. tan = tan =. 751 tan = 5 tan 1 = tan88 = tan = Let s see some problems like the HW! E. 5: Use the tangent ratio to find the value of each variable. Round to the nearest tenth

24 tan 6 = 1 tan 1 = 8 5 Section 8.5 ~ Angles of Elevation & Depression! L.T.: Be able to use the trig ratios with angles of elevation and depression! Angle of Depression: 1 Angle of Elevation: Recall: Alternate interior angles are. So, the angle of depression the angle of elevation.

25 Let s practice! E. 1: Describe each angle as it relates to the picture shown. 1 E. : The angle of depression from the Goodear Blimp to home plate at Busch Stadium is 65. If the ground distance from the blimp to the plate is 900 feet, what is the altitude of the blimp? 65 Angles of depression: ft Angles of elevation: Would it be oka if we just do the homework now? 1. A slide has an angle of elevation of 5. It is 60 feet from the end of the slide to the stairwa beneath the top of the slide. How long is the slide? 5 60 ft. A forester is standing 150 feet awa from a tree. She measures the angle of elevation from where she is to the top of the tree to be 0. How tall is the tree? ft 5

26 . A moving sidewalk takes zoo visitors up a hill. The vertical distance of the sidewalk is 8 feet. Its angle of elevation is 15. About how long is the sidewalk? 15 8 ft 6. An airplane fling 000 feet above ground begins a descent (angle of depression) to land at an airport. How man miles from the airport is the plane when it starts its descent? (Hint: 1 mile = 580 feet) 000 ft. 7. A 100-foot-tall lighthouse stands at the top of a 150-foot-tall cliff. The angle of depression from the top of the lighthouse to a ship is 7. About how far from the cliff is the ship? 50 ft A building is 50 feet high. At a distance awa from the building, an observer notices that the angle of elevation to the top of the building is 1º. How far is the observer from the base of the building? 100 ft 150 ft 1 50 ft 6

27 11. An airplane is fling at a height of miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the airport? mi 5 mi 1. A campsite is 9.1 miles from a point directl below the mountain top. If the angle of elevation is 1 from the camp to the top of the mountain, how high is the mountain? mi 8.6a The Law of Sines! L.T.: Be able to use the Law of Sines to find unknowns in triangles! Quick Review: What does Soh-Cah-Toa stand for? What kind of triangles do we use this for? 7

28 The Law of Sines: sin A a Note: sin B = b capital letters alwas stand for! lower-case letters alwas stand for! sin C = c A c Use the Law of Sines ONLY when: ou DON T have a right triangle AND ou know an angle and its opposite side b B a C Let s do some problems! E. 1: Use the Law of Sines to find each missing angle or side. Round an decimal answers to the nearest tenth. 9 6 a A 79 8 C 8

29 E. : Use the Law of Sines to find each missing angle or side. Round an decimal answers to the nearest tenth. T sin A a sin B b sin C c = = 51 s r E. : Draw ABC and mark it with the given information. Solve the triangle. Round an decimal answers to the nearest tenth. a. = 7, m A = 7, m B = 76 a c 76 7 A 7 b B C 9

30 b. a = 1, m A = 70, c =. 1.1 B 1 A 70 b C The Law of Cosines! L.T.: Be able to use the Law of Cosines to find unknowns in triangles! B c a Note: capital letters alwas stand for! lower-case letters alwas stand for! A C b Use the Law of Cosines ONLY when: ou DON T have a right triangle AND ou can t use the Law of Sines 0

31 Let s do some problems! E. 1: Use the Law of Cosines to solve each triangle. Round an decimal answers to the nearest tenth. a. In ABC, m A = 5, b = 16, and c = B A 5 16 a C b. In CAT, a = 16, t = 0, and m C = 0. A c 0 0 T 16 C 1

32 c. In RED, r = 8, e = 0, and d = 16. R 16 E 0 8 D