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 Similarity in Right Triangles Objectives Determine the geometric mean between two numbers. State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.

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 “x” is the geometric mean between “a” and “b” if: x 2 = ab √x 2 = √ab x = +/- √ab 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?

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

Simplifying Radical Expressions (pg. 287) No “party people” under the radical No fractions under the radical No radicals in the denominator Party People are perfect square #’s which are?

White Board Practice Simplify

White Board Practice Simplify

White Board Practice Simplify

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

White Board Practice Simplify

Warm-up Simplify Find Geometric Mean of 7 and 12

White Board Practice Simplify

Similarity and Geometric Mean Similar Triangle Example What is special about a geometric mean proportion? We are now going to combine the idea of similarity with a geometric mean proportion.

SHMOOP VID meanhttp:// mean

Theorem altitude If the altitude is drawn to the hypotenuse of a right triangle….. –2 additional right triangles are created –The 3 triangles are all similar Their sides are in proportion to one another b y g o p Note: What one color side represents to one triangle, represents something different in another!

Hypotenuse Big LegSmall Leg OG Triangle Medium Small Fill in the table with the letter of the color that represents each part of each different triangle. PARTNERS: Find all of similarity proportions that would create geometric mean problems.

Corollary altitude When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments on the hypotenuse. b y g o p Easier way to remember… create the proportion of the legs of both smaller triangles.

Corollary When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg (closest to that leg.) b y g o p

Group Practice Pg. 288 #17 a. √14 b. 3√ 2 c. 3 √ 7

Group Practice If RS = 2 and SQ = 8 find PS PS = 4 R P Q S

Group Practice If RP = 10 and RS = 5 find RQ RQ = 20 R P Q S

Group Practice If RS = 4 and PS = 6, find SQ SQ = 9 R P Q S

8.2 The Pythagorean Theorem Objectives State and apply the Pythagorean Theorem. Examine proofs of the Pythagorean Theorem.

WARM - UP Label the triangle with 4 letters Re-draw the 3 similar triangles, lining them up so that their corresponding parts are in the same position Write down 1 of the 3 proportions that create a geometric mean

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.”

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? Identify the error or errors in their version of this well-known result. Is their statement true for any triangles at all? If so, which ones?

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. Use complete sentences.

The Pythagorean Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. a b c Brightstorm - proof

Find the value of each variable 1. x 3 2

Find the value of each variable y

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

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

Find the value of each variable 3. 4 x x

Find the value of each variable 5. 4 X + 2 x

Find the value of each variable 5. X 2 + (x+2) 2 = 10 X 2 + x 2 + 4x + 4 = 100 2x 2 + 4x – 96 = 0 X 2 + 2x – 48 = 0 (x + 8)(x – 6) = 0 X = -8 ; x = 6 10 X + 2 x

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 c 2 a 2 + b 2 –c always represents the longest side Lets try… what type of triangle has sides lengths of 3, 4, and 5?

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. a b c Right Triangle

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, 138, 15, 177, 24, 25 6,8,1010,24,26 9,12,15 12,16,20 15,20,25 This column should be memorized!!

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. a b c Triangle is acute a= 6, b = 7, c = 8 Is it a right triangle?

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 b c Triangle is obtuse a= 3, b = 7, c = 9 Is it a right triangle?

Review We use c 2 a 2 + b 2 C 2 = then we a right triangle C 2 < then we have acute triangle C 2 > 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. acute

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?

WARM - UP Solve for x, y, and z x y z 16 4

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

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

The sides opposite the 45 ◦ angles are congruent. The side opposite the 90 ◦ angle is the length of the leg multiplied by √2 Look for the pattern.. USE THIS SET UP EVERY TIME YOU HAVE ONE OF THESE PROBLEMS!!!

Look for the pattern

10

Look for the pattern 10

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

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

30º-60º-90º Theorem x 2x Hypotenuse = 2 ∙ short leg Long leg = √3 ∙ short leg x

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

Look for the pattern

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

White Board Practice 9 y x 60º 30º 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 Geometric mean / simplifying radical expressions Corollary 1 & 2 - ** #32 p. 289 ** 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 – triangles (problems using squares) triangles (problems using equilateral triangles )

WARM-UP What is the one piece of information we need to prove 2 RIGHT triangles are similar? Explain in complete sentences why.

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

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

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

The Tangent Ratio The tangent of an acute angle is defined as the ratio of the length of the opposite leg divided by the adjacent leg of the right triangle. Tangent L A = Tan A length of opposite leg length of adjacent leg C opposite Adjacent A B

Find Tan A A B C 7 2 Tan A

Find Tan B A B C 7 2 Tan B

How do we use it? 1.If we know the ratio we can use it to determine the measurement of the angle –We either look up the value of the ratio in the book on page 311 –Or we use a scientific calculator by entering the ratio and then pressing inverse TAN (TAN -1 )

Find  A A B C 7 2 B Tan A Tan A ≈ pg (TAN -1 )

 A A B C 7 2

Find  B A B C 7 2 

 A A B C 17 8 Find  A

How do we use it? 2.If we know the angle degree measure we can use it to find a missing side length –Look it up in the table (pg. 311) by finding the degree and then looking under Tangent –Or we use scientific calculator by entering the degree measure and then pressing TAN

Find the value of x to the nearest tenth 35º 10 x Tan 35º.7002

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

Find the measure of angle y yºyº 8 5

Page 306 #7 Brightstorm

Find the value of x to the nearest tenth X 20 24º

Find the measurement of angle x XºXº

ON PG. 311… WHY IS THE TANGENT RATIO FOR 45 ◦ 1.000? WHY IS THE TANGENT RATIO FOR 60 ◦ ? WARM-UP

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 The cosine of an acute angle is defined as the ratio of the length of the adjacent leg to the hypotenuse of the right triangle. Cosine L A = Cos A length of adjacent leg length of hypotenuse C opposite Adjacent A B Hypotenuse

Find Cos A A B C Cos A 9

A B C Cos A 9  A ≈ 53 ▫ cos A ≈.6 - pg (COS -1 ) Find  A

The Sine Ratio The sine of an acute angle is defined as the ratio of the length of the opposite leg to the hypotenuse of the right triangle. Sine L A = sin A length of opposite leg length of hypotenuse C opposite Adjacent A B Hypotenuse

Find Sin A A B C Sin A 9

sin A  A ≈ 53 ▫ sin A ≈.8 - pg (SIN -1 ) Find  A using sine A B C

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.

With a partner try to come up with a new saying. S O H C A H T O A

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.

Whiteboards Page 313 –#7, 9

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

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

Brightstorm

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 Angle of depression = Angle of elevation 2º2º 2º2º Horizontal

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) 2º2º 2º2º Horizontal 25m X ≈ 716m 88º

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 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. 40º 80 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!!

Example 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? 38º 3km x

/basic-trigonometry/angle-of-elevation-and- depression-problem-1/ /basic-trigonometry/angle-of-elevation-and- depression-problem-1/

Grade Incline of a driveway or a road Grade = Tangent

Example A driveway has a 15% grade –What is the angle of elevation? xºxº

Example Tan = 15% Tan xº =.15 xºxº

Example Tan = 15% Tan xº =.15 9º9º

Example If the driveway is 12m long, about how much does it rise? 9º9º 12 x

Example If the driveway is 12m long, about how much does it rise? 9º9º