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Chapter 8 Right Triangles • Determine the geometric mean between two numbers. • State and apply the Pythagorean Theorem. • Determine the ratios of the sides of the special right triangles. • Apply the basic trigonometric ratios to solve problems. 8.1 Radicals and Geometric Mean Objective • Determine the geometric mean between two numbers. Simplifying Radical Expressions (Complete Page 280 1- 28) • If you are “perfect” you can’t be “rad”!!! • No fractions under the radical 4 4 2 3 3 3 • No radicals in the denominator 2 3 2 3 3 3 3 PROPORTIONS… HOW DO WE SOLVE THEM? 5 8 x = 10 50 = 8x The Geometric Mean “x” is the geometric mean between “a” and “b” if: a x x b Take Notice: The term said to be the geometric mean will always be crossmultiplied w/ itself. Take Notice: In a geometric mean problem, there are only 3 variables to account for, instead of four. 2 x = ab 2 √x = √ab or x ab Example What is the geometric mean between 3 and 6? 3 x x 6 or x 3 6 18 3 2 You try it • Find the geometric mean between 2 and 18. 6 • Complete geo mean problems from workbook Find the Geometric Mean • 2 and 3 – √6 • 2 and 6 – 2√3 • 4 and 25 – 10 Warm-up • Simplify 45 5 (2 3 ) 2 • Find Geometric Mean of 7 and 12 8.2 The Pythagorean Theorem Objectives • State and apply the Pythagorean Theorem. • Examine proofs of the Pythagorean Theorem. Movie Time • We consider the scene from the 1939 film The Wizard Of Oz in which the Scarecrow receives his “brain,” Scarecrow: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” • Write this down as it is shown… • We also consider the introductory scene from the episode of The Simpsons in which Homer finds a pair of eyeglasses in a public restroom… Homer: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” Man in bathroom stall: “That's a right triangle, you idiot!” Homer: “D'oh!” • Homer's recitation is the same as the Scarecrow's, although Homer receives a response Think – Pair - Share 1. What are Homer and the Scarecrow attempting to recite? • • Is their statement true for any triangles at all? If so, which ones? Identify the error or errors in their version of this well-known result. Think – Pair - Share 2. Is the correction from the man in the stall sufficient? • • Give a complete, correct statement of what Homer and the Scarecrow are trying to recite. Do this first using only English words… • and a second time using mathematical notation. The Pythagorean Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. c a b 2 2 2 c a b Proof demo - cutout • Complete pyth th worksheet in workbook Find the value of each variable 1. x 13 x 2 3 Find the value of each variable 2. y 2 5 y 4 6 Find the length of a diagonal of a rectangle with length 8 and width 4. 4. 8 4 4 8 Find the length of a diagonal of a rectangle with length 8 and width 4. 4. 4 5 4 8 3. Find the length of the diagonal of a square with a perimeter of 20 4. Find the length of the altitude to the base of an isosceles triangle with sides of 5, 5, 8 Warm – up • Create a diagram and label it… • An isosceles triangle has a perimeter of 38in with a base length of 10 in. The altitude to the base has a length of 12in. What are the dimensions of the right triangles within the larger isosceles triangle? 8.3 The Converse of the Pythagorean Theorem Objectives • Use the lengths of the sides of a triangle to determine the kind of triangle. • Determine several sets of Pythagorean numbers. Given the side lengths of a triangle…. • Can we tell what type of triangle we have? YES!! • How? – We use c2 a2 + b2 – c always represents the longest side • Lets try… what type of triangle has sides lengths of 3, 4, and 5? Pythagorean Sets • A set of numbers is considered to be Pythagorean set if they satisfy the Pythagorean Theorem. WHAT DO I MEAN BY SATISFY THE PYTHAGOREAN THEOREM? 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 6,8,10 10,24,26 9,12,15 This column should 12,16,20 be memorized!! 15,20,25 Theorem If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. c a b 2 c a b 2 Right Triangle 2 Theorem (pg. 296) If the square of one side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle. c a= 6 , b = 7, c = 8 Is it a right triangle? a c a b 2 2 2 b Triangle is acute Theorem (pg. 296) If the square of one side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle. a= 3 , b = 7, c = 9 Is it a right triangle? c a c a b 2 b 2 2 Triangle is obtuse Review • We use c2 a2 + b2 2 •C = then we a right triangle 2 •C < then we have acute triangle 2 •C > then we have obtuse triangle • Always make ‘c’ the largest number!! The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 1. 20, 21, 29 • right The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 2. 5, 12, 14 • obtuse The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 3. 6, 7, 8 • acute The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 4. 1, 4, 6 – Not possible The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 5. 3, 4, 5 • acute Warm-up • Solve for x x 2 7 x 3 9 8.4 Special Right Triangles Objectives • Use the ratios of the sides of special right triangles 45º-45º-90º Theorem In a 45-45-90 triangle, the hypotenuse is 2 a leg. times the length of each l = length of leg 45 x√2 x 45 x leg 45º : leg 45º : hypot 90º : 2 Look for the pattern.. USE THIS SET UP EVERY TIME YOU HAVE ONE OF THESE PROBLEMS!!! • The legs opposite the 45◦ angles are congruent. • Hypotenuse opposite the 90◦ angle is the length of the leg multiplied by √2 45º : 45º : 90º : 2 Look for the pattern.. USE PATTERN LIKE ITS AN ALGEBRA PROBLEM leg 45º : 6 leg 45º : hypot 90º : 2 Look for the pattern 45º : 6 45º : 6 90º : 2 6 2 Look for the pattern 45º : 45º : 90º : 2 10 Look for the pattern 45º : 5 2 45º : 5 2 10 90º : 2 White Board Practice 6 x x3 2 x Partner Discussion • If we know the length of a diagonal of a square, can we determine the length of a side? If so, how? x x√2 x White Board Practice • If the length of a diagonal of a square is 4cm long, what is the perimeter of the square? •Perimeter = 8√2cm White Board Practice • A square has a perimeter of 20cm, what is the length of each diagonal? • Diagonal = 5√2 cm 30º-60º-90º Triangle 30 30 60 A 30º-60º-90º triangle is half an equilateral triangle 60 30º-60º-90º Theorem short leg - 30º : big leg - 60º : 3 hypot - 90º : 2 60 THE MEASUREMENTS OF THE PATTERN ARE BASED ON THE 2l l l LENGTH OF THE SHORT LEG ( ) (OPPOSITE THE 30 DEGREE ANGLE) 30 l 3 Look for the pattern.. USE THIS SET UP EVERY TIME YOU HAVE ONE OF THESE PROBLEMS!!! Short leg 30º : Big leg 60º : 3 hypotenuse 90º : 2 Look for the pattern 30º : 6 60º : 3 90º : 2 Look for the pattern 30º : 6 60º : 3 6 3 90º : 2 12 Look for the pattern 30º : 60º : 3 9 90º : 2 Look for the pattern 30º : 3 3 60º : 3 9 90º : 2 6 3 White Board Practice Big leg 5 hypot 60º bigleg 5 3 hypot 10 White Board Practice 9 30º y x 60º y = 3√3 x = 6√3 White Board Practice • Find the length of an altitude of a equilateral triangle if the side lengths are 16cm. •8√3 cm Quiz Review Sec. 1 - 4 8.1 • Geometric mean / simplifying radical expressions 8.2 • Pythag. Thm – rectangle problems - pg. 292 #10, 13, 14 – Isosceles triangle problems pg. 304 #7 8.3 • Use side lengths to determine the type of triangle (right, obtuse, acute) – Pg. 297 1 – 5 8.4 • 45-45-90 triangles (problems using squares) • 30-60-90 triangles (problems using equilateral triangles ) WARM-UP • Proving 2 triangles similar…. We had 3 shortcuts. – Which was the shortest of shortcuts? – AA • If you have 2 right triangles what is the only other piece of info you would need to say they’re similar? Trigonometry Objectives • Understand the basics of trig and the 3 ratios that relate Show music vid Trigonometry basics Pg. 311 • If 2 right triangles have the same acute angle they have to be similar, therefore the ratio of their sides has to be equivalent • Mathematicians have discovered ratios that exist for every degree from 1 to 89. • The ratios exist, no matter what size the triangle 30 30 Trigonometry basics “Triangle measurement” Sides are named relative to an acute angle. Opposite leg B C A Adjacent leg Trigonometry basics Adjacent leg B C Sides are named relative to the acute angle. What never changes? A Opposite leg The Tangent Ratio Tangent LA = length of opposite leg length of adjacent leg B opposite Tan A C Adjacent A Opp Adj Sine and Cosine Ratios • Both of these ratios involve the length of the hypotenuse The Cosine Ratio Cosine LA = length of adjacent leg length of hypotenuse B opposite Cos A C Adjacent A Adj Hyp The Sine Ratio Sine LA = length of opposite leg length of hypotenuse sin A opposite B C Adjacent A opp Hyp • Complete workbook page 21 • Start problems on page 22 Find Tan A A 2 Tan A 7 7 C 2 B Find Tan B 7 Tan B 2 Page 306 Learning to use the trig table and/or you calculator #7 How do we use it? 1. We use the ratio to determine the measurement of the angle – page 311 – (TAN-1) Find m A A WHAT ELEMENTS OF THE TRINALGE TO WE HAVE IN RELATION TO THE A? 2 Tan A 7 7 C 2 B 16 Tan A ≈ .2857 - pg. 311 -.2857 (TAN-1) Find m B A 7 C B 74 2 B How do we use it? 2. Use the measure of the angle to find a missing side length – page 311 – TAN Find the value of x to the nearest tenth 10 35º x x Tan 35º 10 x .7002 10 x 7.0 WHITEBOARDS Find the value of x to the nearest tenth x 78.1 30 21º x WHITEBOARDS Find the measure of angle y 8 5 yº y 58 Find the value of x to the nearest tenth x 8.9 X 20 24º 8.6 The Sine and Cosine Ratios Objectives • Define the sine and cosine ratio Sine and Cosine Ratios • Both of these ratios involve the length of the hypotenuse The Cosine Ratio Cosine LA = length of adjacent leg length of hypotenuse B opposite Cos A C Adjacent A Adj Hyp The Sine Ratio Sine LA = length of opposite leg length of hypotenuse sin A opposite B C Adjacent A opp Hyp Find Cos A A 9 C 9 Cos A 15 15 12 B Find Sin A 12 B C 15 9 A 12 Sin A 15 Using the trig table • Pg. 313 #7 Find m A – set up using COS and SIN A 9 C 15 12 B 9 Cos A 15 cos A ≈ .6 - pg. 311 -.3 (COS-1) 12 sin A 15 sin A ≈ .8 - pg. 311 -.3 (SIN-1) A ≈ 53▫ • Page 313 – 9 and 10 SOH-CAH-TOA Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse Tangent Opposite Adjacent • Some Old Horse Caught Another Horse Taking Oats Away. • Sally Often Hears Cats Answer Her Telephone on Afternoons • Sally Owns Horrible Cats And Hits Them On Accident. So which one do I use? • Sin • Cos • Tan Label your sides and see which ratio you can use. Sometimes you can use more than one, so just choose one. Find the measures of the missing sides x and y x 23º y x ≈ 110 y ≈ 47 100 67º White boards - Example 2 • Find xº correct to the nearest degree. x ≈ 37º xº 18 30 Find the measurement of angle x x 37 6 8 Xº 10 White Board • An isosceles triangle has sides 8, 8, and 6. Find the length of the altitude from angle C to side AB. • √55 ≈ 7.4 8.7 Applications of Right Triangle Trigonometry Objectives • Apply the trigonometric ratios to solve problems • Every problem involves a diagram of a right triangle An operator at the top of a lighthouse sees a sailboat with an angle of depression of 2º Angle of depression = Angle of elevation Horizontal Angle of depression 2º 2º Angle of elevation Horizontal TEMPLATE ANGLE OF ELEVATION / DEPRESSION An operator at the top of a lighthouse (25m) sees a Sailboat with an angle of depression of 2º. How far away is the boat? Horizontal Distance to light house (X) 2º X ≈ 716m 25m 88º 25m 88º 2º Distance to light house (X) • Go to workbook page 29 Example 1 • You are flying a kite is flying at an angle of elevation of 40º. All 80 m of string have been let out. Ignoring the sag in the string, find the height of the kite to the nearest 10m. • How would I label this diagram using these terms.. • Kite, yourself, height (h) , angle of elev., • 80m x Sin 40 80 x .6428 80 51.4 x WHITE BOARDS • An observer located 3 km from a rocket launch site sees a rocket at an angle of elevation of 38º. How high is the rocket? • Use the right triangle to first correctly label the diagram!! x Tan38 3 x .7813 3 2.34 x Grade • Incline of a driveway or a road • Grade = Tangent Example • A driveway has a 15% grade – What is the angle of elevation? xº Example • Tan = 15% • Tan xº = .15 xº Example • Tan = 15% • Tan xº = .15 9º Example • If the driveway is 12m long, about how much does it rise? 12 9º x Example • If the driveway is 12m long, about how much does it rise? 12 9º 1.8