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.

The Geometric Mean “x” is the geometric mean between “a” and “b” if:

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 No “party people” under the radical No fractions under the radical No radicals in the denominator

Theorem If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. a m b n h 1 2 3

Corollary 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. m h n

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.) m a b n

White Board Practice Simplify

White Board Practice

Simplify

White Board Practice

Simplify

White Board Practice

Simplify

White Board Practice

Simplify

White Board Practice

Simplify

White Board Practice

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

Group Practice PS = 4 R P Q S

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

Group Practice SQ = 15 R P Q S

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

Group Practice SQ = 9 R P Q S

8.2 The Pythagorean Theorem Objectives State and apply the Pythagorean Theorem. Examine two proofs of the Pythagorean Theorem. Determine several sets of Pythagorean numbers.

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 Proof

Pythagorean Sets A set of numbers is considered to be Pythagorean set if they satisfy the Pythagorean Theorem. 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

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 “$pringfield (Or, How I Learned to Stop Worrying and Love Legalized Gambling)” 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 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?

Pair 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?

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 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.

Pair 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.

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.

Find the value of each variable 1. x 3 2

Find the value of each variable 1. x 3 2

Find the value of each variable y

Find the value of each variable y

Find the value of each variable 3. 4 x x

Find the value of each variable 3. 4 x x

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

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

8.3 The Converse of the Pythagorean Theorem Objectives Use the lengths of the sides of a triangle to determine the kind of triangle.

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

Theorem 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

Theorem 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 Sketch

The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 1. 20, 21, 29

The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 1. right

The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 2. 5, 12, 14

The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 2. obtuse

The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 3. 6, 7, 8

The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 3. acute

The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 4. 1, 4, 6

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

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

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 a 45 45º 45º 90º a a a

Look for the pattern

30º-60º-90º Theorem a 2a º 60º 90º a a 2a

Look for the pattern

White Board Practice 6 x x

6 x x

5 y x 60º

White Board Practice 5 y x 60º

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

Trigonometry A B C Opposite side Adjacent side Hypotenuse Sides are named relative to an acute angle.

Trigonometry A B C Opposite side Adjacent side Hypotenuse Sides are named relative to the acute angle.

The Tangent Ratio The tangent of an acute angle is defined as the ratio of the length of the opposite side to the adjacent side of the right triangle that contains the acute angle. Tangent Angle A Tan A

Find Tan A A B C 7 2

Tan A A B C 7 2

Find Tan B A B C 7 2

Tan B A B C 7 2

Find  A A B C 7 2

 A A B C 7 2

Find  B A B C 7 2

 B A B C 7 2

Find Tan A A B C 17 8

Tan A A B C 17 8

Find Tan B A B C 17 8 A B C 8

Tan B A B C 17 8

Find  A A B C 17 8

 A A B C 17 8

Find  B A B C 17 8

 B A B C 17 8

Find the value of x to the nearest tenth 35º 10 x

Find the value of x to the nearest tenth 35º 10 x

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

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

Find the value of x to the nearest tenth yºyº 8 5

yºyº 8 5

yºyº

Find the value of x to the nearest tenth yºyº

8.6 The Sine and Cosine Ratios Objectives Define the sine and cosine ratio

The Cosine Ratio The cosine of an acute angle is defined as the ratio of the length of the adjacent side to the hypotenuse of the right triangle that contains the acute angle. Sketch

The Sine Ratio The sine of an acute angle is defined as the ratio of the length of the opposite side to the hypotenuse of the right triangle that contains the acute angle. Sketch

SOHCAHTOA 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.

Your Turn - Think S O H C A H T O A

Your Turn - Pair SOHCAHTOASOHCAHTOA

Your Turn - Share SOHCAHTOASOHCAHTOA

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.

Example 1 Find the values of x and y to the nearest integer.

Example 2 Find xº correct to the nearest degree.

Example 3 Find the measures of the three angles of  ABC.

Example 4 Find the lengths of the three altitudes of  ABC

8.7 Applications of Right Triangle Trigonometry Objectives Apply the trigonometric ratios to solve problems

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

Example 1 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.

Example 1 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 x

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?

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? 38 3 x

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