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Geometry Name ________________________ Date _______________ Hour _____ Assignment Intro to Ch. 8 Worksheet 8.1 Worksheet 8.2 Day 1 Worksheet 8.2 Day 2 Worksheet 8.3 Day 1 Worksheet 8.3 Day 2 Worksheet Review Review Worksheet Quiz 8.4 Day 1 Worksheet 8.4 Day 2 Pg. 567 #16-33, 36-42 even. 57-59 8.4 Extra Practice Worksheet 8.5 pg 577 # 1-17 odd, 32-37 8.5 In-class project Worksheet (Be ready for a quiz tomorrow) 8.6 Day 1 Worksheet 8.6 Day 2 Worksheet Ch. 8 review Review Worksheet Geometry Chapter 8 Learning Targets! By the end of the chapter, you 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 hypotenuse •Use the Pythagorean Theorem and its converse •Use the properties of 45-45-90 triangles and 30-60-90 triangles •Find trigonometric ratios using right triangles, and use these ratios to find angle measures in right triangles •Identify •Use and use Angles of Depression and Angles of Elevation to solve problems and find missing values the Law of Sines and Law of Cosines to find missing values in triangles 1 Geometry Name ________________________ Date _______________ Hour _____ Ch. 8 Introduction L.T.#1: Be able to write radicals in “simplest radical form”! L.T.#2: Be able to solve equations involving radicals! Quick Vocab: Vocab: What is a radical? What is simplest radical form? Now, let’ let’s fold some socks! 18 150 124 You can multiply numbers that are under radicals! 2⋅ 8 5 3⋅ 8 3 5 ⋅4 3 16 ⋅ 120 2 Geometry Name ________________________ Date _______________ Hour _____ You can divide numbers that are under radicals! But, NEVER leave a radical sign in your denominator! You must RATIONALIZE it!! 294 ÷ 3 4 12 4 3 5 25 2 20 4 5 ⋅ 30 3 Now, let’s solve some equations using radicals! Leave your answers in simplest radical form! x 2 + 4 = 20 2 x 2 = 20 − 4 + x 2 = 140 1 2 x 2 + 4 = 20 3 Geometry Name ________________________ Date _______________ Hour _____ Did we meet the target? L.T.#1: Be able to write radicals in “simplest radical form”! L.T.#2: Be able to solve equations involving radicals! Solve this equation and write your answer in simplest radical form! 2 x 2 − 4 = 36 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 Ex. 1: Identify the geometric mean in each of the following proportions. 3 y GM: 3 4 GM: x 4 GM: = 4 = = x 5 y x 7 Ex. 2: Find the value of the geometric mean. 4 x = x 16 y 4 = 5 y 4 Geometry Name ________________________ Date _______________ Hour _____ Ex. 3: Find the geometric mean of: a. 3 and 12 b. 3 and 48 c. 15 and 20 The Geometric Mean in Right ∆s! Theorem: In a right triangle, when you draw an altitude to the hypotenuse, you create three similar triangles! ∆ACD ~ ∆ADB ~ ∆DCB A C B D The legs and altitude of a right triangle are the geometric means between the segments of the hypotenuse that ______________________________. a b c x y z 5 Geometry Name ________________________ Date _______________ Hour _____ Let’s practice! Ex. 4: Refer to the picture to complete each proportion. x = z y Challenge: b z = a a x = c = b = z b y a z y y b = z b z x y c Ex. 5: Find the values of the variables. 9 x y x 4 2 8 10 6 Geometry Name ________________________ Date _______________ Hour _____ Just a few more! 5 x 9 x z 3 4 y 1 3 Section 8.2 ~ day 1 The Pythagorean Theorem!! L.T.: Be able to find unknowns in triangles using the Pythagorean Theorem! Quick WarmWarm-up: x 2 + 3 = 15 22 + x 2 = 62 What do you call the longest side of a right triangle? (the side opposite the right angle) 7 Geometry Name ________________________ Date _______________ Hour _____ Pythagorean 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 hypotenuse. a2 + b2 = c2 c a b Find the value of x. Leave your answer in simplest radical form. 8 x 4 x 10 x 3 4 4 Pythagorean Triples! A Pythagorean Triple is a set of whole numbers that satisfies the equation a2 + b2 = c2. Common triples: (Multiples of these sets are also triples!) Find the value of x. 12 13 x 8 x 15 6 x 8 8 Geometry Name ________________________ Date _______________ Hour _____ Did we meet the target? L.T.: Be able to find unknowns in triangles using the Pythagorean Theorem! Find the length of the hypotenuse! 9 12 Section 8.2 ~ day 2 The Pythagorean Theorem!! L.T.#1: Be able to find unknowns in triangles using the Pythagorean Theorem, and find the areas of triangles! Quick Review: Find the value of each variable! Leave answers in simplest radical form. 12 12 x y 16 16 What is an example of a Pythagorean Triple? 9 Geometry Name ________________________ Date _______________ Hour _____ Using Pythagorean Theorem to find AREAS! Find the area of the triangle. Do the legs form a triple? 42 m 445 m A right triangle has a hypotenuse of length 25 and a leg of length 10. Find the area of the triangle in simplest radical form and also as a decimal. If a 2 + b 2 < c 2 , then the ∆ is obtuse. If a 2 + b 2 > c 2 , then the ∆ is acute. The lengths of the sides of a triangle are given. Is the triangle right, obtuse, or acute? Explain. 12, 16, 20 11, 12, 15 31, 23, 12 11, 7 , 4 A window-washer leans a 41-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 Geometry Name ________________________ Date _______________ Hour _____ Did we meet the target? L.T.#1: Be able to find unknowns in triangles using the Pythagorean Theorem, and find the areas of triangles! Find the area of the triangle. 26m 24m Section 8.3 ~ day 1 Special Right Triangles! L.T.: Be able to find sides of 45-45-90 triangles! Quick Review: Rationalize the following! 24 6 32 2 6 3 2 11 Geometry Name ________________________ Date _______________ Hour _____ Find the length of the hypotenuse: 3 8 3 8 45-45-90 Triangle Theorem: In a 45-45-90 triangle, both ______ are congruent (isosceles), and the length of the hypotenuse is _____ times the length of each leg. Find the value of each variable: 5 3 x x 5 2 y 45° 4 2 x y 45° y x 45° 21 2 4 12 x 45° 12 Geometry Name ________________________ Date _______________ Hour _____ Find the value of each variable! 7 2 6 x 45° x y 45° Find the area of the triangle! 4 2 6 45° 45° 13 Geometry Name ________________________ Date _______________ Hour _____ Why do we need to know about 45-45-90 triangles? They are in the real world! Not to mention, it’s a whole lot easier than using the Pythagorean Theorem. Yadier Molina wants to know how far he has to throw the ball to catch a man stealing second. 90 ft Before, we had to use the Pythagorean Theorem. But now that you know about 45-45-90 triangles, you can use the shortcut! x ft 90 ft Did we meet the target? L.T.: Be able to find sides of 45-45-90 triangles! Find the value of x. x 14 45° 14 Geometry Name ________________________ Date _______________ Hour _____ Section 8.3 ~ day 2 Special Right Triangles! L.T.: Be able to find sides of 30-60-90 triangles! Quick Review: y 4 2 x 45° 5 45° x y Question: In a 30-60-90 triangle, will any sides be the same length? No! 30-60-90 Triangle Theorem: In a 30-60-90 triangle, the length of the hypotenuse is ____ times the length of the shorter leg (shorty) the length of the longer leg (longy) is _____ times the length of the shorter leg (shorty) “shorty” 60° hyp ote nu se 30° “longy” 15 Geometry Name ________________________ Date _______________ Hour _____ Find the value of each variable! 60° x 3 7 x 30° y y 15 y x 30° 30° 5 12 y 60° x x y 30° x y 3 3 15 y x 30° 30° 5 y x 16 Geometry Name ________________________ Date _______________ Hour _____ Let’s practice some more! Find the area of each triangle. x 60° 6 x 3 3 30° y A window-washer leans a 40-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? 40 ft y 60° x d Find the value of each variable. c 60° b 7 2 45° a 17 Geometry Name ________________________ Date _______________ Hour _____ Did we meet the target? L.T.: Be able to find sides of 30-60-90 triangles! Find the value of each variable! 5 2 a 60° 45° b d c Section 8.4 day 2 ~ 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: The Cosine Ratio: In a right triangle, the ratio of the leg OPPOSITE an angle to the __________________ is a constant. This is called the sine 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! sin ∠ = opp hyp A cos ∠ = C B adj hyp A C B 18 Geometry Name ________________________ Date _______________ Hour _____ Let’s practice! V Ex. 1: Write the sine and cosine ratios for ∠ T. T 5 13 5 V 12 4 T U 3 U Ex. 2: Find each of the following using your calculator! Round decimals to the nearest thousandth. sin 120 = sin 15 = cos 45 = You can find an angle from a given sine or cosine ratio! These are called “inverse sine” and “inverse cosine”and you can use the sin-1 and cos-1 buttons on your calculator! Ex. 3: Find the angle with the given ratio using your calculator! Round decimals to the nearest tenth. cos x = 0.7986 sin x = 0.866 tan x = 5.6713 Ex. 4: Use the triangle to find each of the following. a. sin A = b. cos A = A 5 c. sin B = d. cos B = C 13 12 B 19 Geometry Name ________________________ Date _______________ Hour _____ Let’s practice with the calculators! cos 24 = x sin 41 = x tan x = 2.4751 cos 8 = x sin x = 4 5 cos x = 0.3057 Let’s see if we’ve met the learning target! Ex. 5: Use the appropriate ratio to find the value of each variable. 24° 20 6 x x 35° x° 10 x° 10 6.5 27 20 Geometry Name ________________________ Date _______________ Hour _____ Section 8.4 day2 ~ The Tangent Ratio! L.T.: Be able to use the tangent ratio to find lengths and angles in triangles! Quick Review: How do you find the third side of a right triangle when you already know two sides? What are the two types of “special” right triangles we have worked with? What is trigonometry? Let’s look at a right triangle! A What side is the hypotenuse? What is the leg opposite ∠ B? opposite What is the leg adjacent to ∠ B? C hypotenuse 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! A tan ∠ = opp adj C B 21 Geometry Name ________________________ Date _______________ Hour _____ Let’s practice! Ex. 1: Write the tangent ratios for ∠ T and ∠ U. T 5 13 5 V 4 V 12 U T 3 U Ex. 2: Find the tangent RATIO of each angle using your calculator! Round decimals to the nearest thousandth. tan 120 = tan 45 = tan 30 = 27° You can find an angle from a given tangent ratio! This is called an “inverse tangent” and you can use the tan-1 button on your calculator! Ex. 3: Find the ANGLE with the given tangent ratio using your calculator! Round decimals to the nearest tenth. tan A = 2.21 tan B = −1.482 Ex. 4: Use the triangle to find each of the following. tan A = A b. m ∠ A = 6 a. c. m ∠B = C 15 B 22 Geometry Name ________________________ Date _______________ Hour _____ Let’s practice with the calculators! Round decimals to the nearest tenth. tan 24 = x tan x = tan x = 2.4751 4 5 tan 41 = x tan 88 = x tan x = 0.3057 Let’s see some problems like the HW! Ex. 5: Use the tangent ratio to find the value of each variable. Round to the nearest tenth. 20 8 24° x° 6 x x 35° 10 6 23 Geometry Name ________________________ Date _______________ Hour _____ tan 46 = x 12 tan 12 = 3 x x° 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: Angle of Elevation: 1 2 Recall: Alternate interior angles are _____. So, the angle of depression _____ the angle of elevation. 24 Geometry Name ________________________ Date _______________ Hour _____ Let’s practice! Ex. 1: Describe each angle as it relates to the picture shown. 1 Ex. 2: The angle of depression from the Goodyear 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˚ 2 3 x 4 65˚ 900 ft Angles of depression: Angles of elevation: Would it be okay if we just do the homework now? ☺ 1. A slide has an angle of elevation of 25˚. It is 60 feet from the end of the slide to the stairway beneath the top of the slide. How long is the slide? x 25˚ 60 ft 2. A forester is standing 150 feet away from a tree. She measures the angle of elevation from where she is to the top of the tree to be 30˚. How tall is the tree? x 30˚ 150 ft 25 Geometry Name ________________________ Date _______________ Hour _____ 3. A moving sidewalk takes zoo visitors up a hill. The vertical distance of the sidewalk is 48 feet. Its angle of elevation is 15˚. About how long is the sidewalk? x 48 ft 15˚ 6. An airplane flying 4000 feet above ground begins a 2˚ descent (angle of depression) to land at an airport. How many miles from the airport is the plane when it starts its descent? (Hint: 1 mile = 5280 feet) 2˚ 4000 ft. 2˚ x 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 27˚. About how far from the cliff is the ship? 100 ft 250 ft 150 ft 27˚ x 10. A building is 50 feet high. At a distance away from the building, an observer notices that the angle of elevation to the top of the building is 41º. How far is the observer from the base of the building? 50 ft 41˚ x 26 Geometry Name ________________________ Date _______________ Hour _____ 11. An airplane is flying at a height of 2 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? 2 mi 5 mi x˚ 13. A campsite is 9.41 miles from a point directly below the mountain top. If the angle of elevation is 12° from the camp to the top of the mountain, how high is the mountain? x 12˚ 9.41 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? 27 Geometry Name ________________________ Date _______________ Hour _____ The Law of Sines: B sin A sin B sin C = = a b c c A a b C Note: capital letters always stand for __________! lower-case letters always stand for ________! Use the Law of Sines ONLY when: you DON’T have a right triangle AND you know an angle and its opposite side Let’s do some problems! ☺ Ex. 1: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth. 63° a 29 A 79˚ 38˚ 42 C 28 Geometry Name ________________________ Date _______________ Hour _____ Ex. 2: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth. T sin A sin B sin C = = a b c 51˚ r s 89° 40° 4.8 Ex. 3: Draw ∆ABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth. B a. a = 7, m∠A = 37, m∠B = 76 c A 76˚ 7 37˚ b C 29 Geometry Name ________________________ Date _______________ Hour _____ b. a = 12, m∠A = 70, c = 3.1 B 3.1 A 12 70˚ C b The Law of Cosines! L.T.: Be able to use the Law of Cosines to find unknowns in triangles! B c A a b C Note: capital letters always stand for __________! lower-case letters always stand for ________! Use the Law of Cosines ONLY when: you DON’T have a right triangle AND you can’t use the Law of Sines 30 Geometry Name ________________________ Date _______________ Hour _____ Let’s do some problems! ☺ Ex. 1: Use the Law of Cosines to solve each triangle. Round any decimal B answers to the nearest tenth. a. 19 In ∆ABC , m∠A = 35, b = 16, and c = 19. A 35˚ a 16 C b. In ∆CAT , a = 16, t = 20, and m∠C = 40. A c T 16 20 40˚ C 31 Geometry Name ________________________ Date _______________ Hour _____ c. In ∆RED, r = 8, e = 20, and d = 16. 16 E R 8 20 D 32