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 3n) No “party people” under the radical No fractions under the radical No radicals in the denominator Party People are perfect square #’s which are?   These are the three “nevers” for simplifying a radical. Do additional examples as you see fit.

Means-Extremes property of proportions The product of the extremes equals the product of the means. = a b c d ad = cb

The Geometric Mean x2 = ab √x2 = √ab “x” is the geometric mean between “a” and “b” if: x2 = ab √x2 = √ab The geometric mean comes from a special proportion. It is useful to compare it to the arithmetic mean (average) and show that it is a number between two other numbers. Either definition above can work, but each serves a purpose, and can be used to solve different problems. Take Notice: The term said to be the geometric mean will always be cross-multiplied w/ itself. Take Notice: In a geometric mean problem, there are only 3 variables to account for, instead of four.

Example What is the geometric mean between 3 and 6? This example shows that we have to be able to simplify radical expressions if we want to solve geometric mean problems. Most of the problems involving right triangles require us to simplify redical expressions. Lets review the rules now.

You try it Find the geometric mean between 2 and 18. 6

Find the Geometric Mean 2 and 3 √6 2 and 6 2√3 4 and 25 10

Warm-up Simplify 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 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 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. While most students know or have heard of the Pythagorean theorem, most do not know why it is true. Many proofs have been done (set of 36 different proofs at http://cut-the-knot.com/pythagoras/index.html) and you should demonstrate your two favorite ones to them. c a b Brightstorm - proof

Find the value of each variable 1. x 2 3

Find the value of each variable 2. y 4 6

Find the length of a diagonal of a rectangle with length 8 and width 4.

Find the length of a diagonal of a rectangle with length 8 and width 4.

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?

Theorem Right Triangle 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. Right Triangle This is the converse to the Pythagorean theorem. Any triangle whose sides satisfy these conditions is a right triangle. c a b

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 12,16,20 15,20,25 Although many more sets exist, we should only commit a few to memory. This column should be memorized!!

Theorem (pg. 296) Triangle is acute 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. a= 6 , b = 7, c = 8 Is it a right triangle? c See note on next slide. a b Triangle is acute

Theorem (pg. 296) Triangle is obtuse 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? Bundle the three previous theorems together and you get an all-purpose triangle determination tool. If the length of the longest side works with the other two sides, then the triangle is right. If the length of the longest side is too short, then the triangle is acute, and if it is too long, it is obtuse. Use the sketch to demonstrate this. c a b Triangle is obtuse

C2 = then we a right triangle C2 < then we have acute triangle Review We use c2 a2 + b2 C2 = then we a right triangle C2 < then we have acute triangle C2 > then we have obtuse triangle Always make ‘c’ the largest number!!

The sides of a triangle have the lengths given The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 20, 21, 29 right

The sides of a triangle have the lengths given The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 5, 12, 14 obtuse

The sides of a triangle have the lengths given The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 6, 7, 8 acute

The sides of a triangle have the lengths given The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 1, 4, 6 Not possible

The sides of a triangle have the lengths given The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 5. acute

8.4 Special Right Triangles Objectives Use the ratios of the sides of special right triangles

45º-45º-90º Theorem a Hypotenuse = √2 ∙ leg 45 x√2 x 45 x

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

Look for the pattern.. USE PATTERN LIKE ITS AN ALGEBRA PROBLEM

Look for the pattern 42

Look for the pattern 10 43

Look for the pattern 10 44

White Board Practice Hypotenuse = √2 * leg 6 = √2 x 6 x 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√2 x x

Perimeter = 8√2cm 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

Diagonal = 5√2 cm White Board Practice A square has a perimeter of 20cm, what is the length of each diagonal? Diagonal = 5√2 cm

A 30º-60º-90º triangle is half an equilateral triangle

30º-60º-90º Theorem THE MEASUREMENTS OF THE PATTERN ARE BASED ON THE LENGTH OF THE SHORT LEG (OPPOSITE THE 30 DEGREE ANGLE) 60 2x x 30 x

Look for the pattern.. USE THIS SET UP EVERY TIME YOU HAVE ONE OF THESE PROBLEMS!!! Short leg Long leg hypotenuse

Look for the pattern

Look for the pattern

Look for the pattern On this I would want to solve for a by dividing by rad 3. Once I know a, I can solve for the hypotenuse

Look for the pattern

White Board Practice x 5 Hypotenuse = 2 ∙ short leg y Long leg = √3 ∙ short leg 60º

White Board Practice 9 30º y = 3√3 x = 6√3 y x 60º

8√3 cm 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. AA, SAS, SSS What is the one additional piece of information we need to prove 2 RIGHT triangles are similar? (look at the shortcuts above)

8.5 The Tangent Ratio Objectives Define the tangent ratio for a right triangle Show music vid

Trigonometry Pg. 311 When you have a right triangle you always have a 90◦ angle and 2 acute angles Based on the measurements of those acute angles you can discover the lengths of the sides of the right triangle Mathematicians have discovered ratios that exist for every degree from 1 to 89. The ratios exist, no matter what size the triangle

Trigonometry “Triangle measurement” B Sides are named relative to an acute angle. Hypotenuse Opposite leg A C Adjacent leg

Trigonometry What never changes? Sides are named relative to the acute angle. B What never changes? Hypotenuse Adjacent leg A C Opposite leg

The Tangent Ratio Tangent LA = Tan A length of opposite leg length of adjacent leg Tangent LA = B Tan A Show both how to use the tables on page 311 as well as a calculator. C opposite A Adjacent

Find Tan A A Tan A 7 Find Tan B B C 2 Tan B

Page 306 Learning to use the trig table and/or you calculator #7

How do we use it? We use the ratio to determine the measurement of the angle page 311 (TAN-1)

Find m A WHAT ELEMENTS OF THE TRINALGE TO WE HAVE IN RELATION TO THE  A? Tan A 7 Tan A ≈ .2857 - pg. 311 -.2857 (TAN-1) B B C 2

Find m B A 7  B C 2

How do we use it? 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 Tan 35º .7002

WHITEBOARDS Find the value of x to the nearest tenth 30 21º x

WHITEBOARDS Find the measure of angle y 8 5 yº

Find the value of x to the nearest tenth 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 = Cos A length of adjacent leg length of hypotenuse Cosine LA = B Cos A Show both how to use the tables on page 311 as well as a calculator. C opposite Hypotenuse A Adjacent

The Sine Ratio Sine LA = sin A length of opposite leg length of hypotenuse Sine LA = sin A B Show both how to use the tables on page 311 as well as a calculator. C opposite Hypotenuse A Adjacent

Find Cos A A 15 9 Cos A B C 12

Find Sin A 12 B C 9 Sin A 15 A

Using the trig table Pg. 313 #7

Find m A – set up using COS and SIN - pg. 311 -.3 (COS-1) Cos A A sin A ≈ .8 - pg. 311 -.3 (SIN-1) sin A 15 9 B  A ≈ 53▫ C 12

Page 313 9 and 10

SOH-CAH-TOA Sine Opposite Hypotenuse Cosine Adjacent Tangent

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 23º y 100 x ≈ 110 y ≈ 47 67º

x ≈ 37º White boards - Example 2 Find xº correct to the nearest degree. x ≈ 37º xº 18 30

Find the measurement of angle x 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 2º Angle of depression Angle of elevation 2º Horizontal

TEMPLATE ANGLE OF ELEVATION / DEPRESSION

X ≈ 716m 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? Distance to light house (X) Horizontal 2º X ≈ 716m 88º 25m 25m 88º 2º Distance to light house (X)

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

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!!

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 x 9º

Example If the driveway is 12m long, about how much does it rise? 12 1.8 9º