# Geometry 7-1 Geometric Mean and the Pythagorean Theorem

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Geometry 7-1 Geometric Mean and the Pythagorean Theorem A. Geometric Mean

1. Def: The geometric mean between two positive numbers a and b is the positive number x where: a x

x = . b

Ex 1: Find the geometric mean between the \$8,000 question and the \$32,000 question on “Who Wants to be a Millionaire?”.

Ex 2: Find the geometric mean between 2 and 10.

B. Theorem 7-1

If the altitude is drawn from the vertex of the right angle of a triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other.

___ ___

ABC A B

V : V : V

C. Theorem 7-2

The measures of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is, the geometric mean between the

measures of the two segments of the hypotenuse.

ADB BDC V : V AD BD DB = DC w a a = x Ex 3: Find the length of the altitude, if the following is true.

A B C D C A B D a w x 6 20

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D. Theorem 7-3

If the altitude is drawn to the hypotenuse of a right triangle, then the measure of a leg of the triangle is the geometric mean between the

measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg. ABC ADB BDC V : V : V AD AB AB = AC ( ) w b b = x w hyp+ Ex 4: Find the length of the given sides if the following is true.

HW: Geometry 7-1 p. 346-348

13-32 all, 35-38 all, 42-43, 49-50, 55-65 odd Hon: 34, 44, A B C D a w x b

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Geometry 7-2 The Pythagorean Theorem and its Converse A. Theorem 7-4 - Pythagorean Theorem

In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.

a2+b2 = c2

Ex 1: Find x.

B. Theorem 7-5 - Converse of the Pythagorean Theorem

-If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

1. A Pythagorean Triple is ___________ whole numbers that satisfy the equation a2+b2 = . c2

Ex 2: Determine if the measures of these sides are the sides of a right triangle. 40, 41, 48

HW: Geometry 7-2 p. 354-356

13-29 odd, 30-35, 40, 46-47, 51-55 odd, 61-69 odd Hon: 39

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Geometry 7-3 Special Right Triangles A. 45o−45 90o− o Triangles

Do the Pythagorean Theorem (solve for d)

2 2 2

a +b = c

2 2 2

x +x = d

1. Theorem 7-6 - In a 45 45 90o− o− o triangle,

the length of the hypotenuse is 2 times as long as a leg.

Example 1: Find the length of the sides of the triangle.

Example 2: If the leg of a 45o−45 90o− o triangle is 12 units, find the length

of the hypotenuse. B. 30o−60 90o− o Triangles

1. What is the relationship between the short leg of a 30o−60 90o− o, triangle and the hypotenuse?

short leg (___) = hypotenuse

2. Let’s do Pythagorean Theorem to solve for a.

2 2 (2 )2

a +x = x

short leg (____) = long leg x x d leg 2 hypotenuse= s s s 2 6 A B C D 60o 60o x x a

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1. Theorem 7-7 - In a 30 60 90o− o− o triangle, the

length of the hypotenuse is twice the length of the short leg, and the length of the long leg is 3 times the length of the short leg.

Example 3: Find AB and AC.

Example 4: VWXY is a

30o−60 90o− o triangle with right

angle X and WX as the longer leg. Graph points X (-2, 7) and Y(-7, 7), and locate point W in quadrant III. HW: Geometry 7-3 p. 360-362 12-25, 27, 29, 36, 37, 40, 43-44, 45-65 odd Hon: 26, 38 3 n n 2n 60o 30o 60o 12 A B C

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7-4 Trigonometry Ratios in Right Triangles A. Ratios

1. Trigonometry helps us solve measures in right triangles.

a. Trigon means triangle b. Metron means measure B. Triangle measures

Abbreviation Definition

sin A leg opposite A a

hypotenuse∠ = c

cos A leg adjacent to A b

hypotenuse∠ = c

tan A leg opposite to A a

adjacent ∠ = b Example 1: Find the sin S, cos S, tan S, sin E, cos E, tan E.

Ex 2: Solve the triangle B C A a b c 6 35o C B A M E S 6 10 8

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Ex. 3: Solve the triangle

Ex 4: A plane is one mile above sea level when it begins to climb at a constant angle of 2o for the next 70 ground miles. How far above sea level is the plane

after its climb?

HW: Geometry 7-4 p. 368-370 18-48, 63-64, 69-81 odd Hon: 55-58, 65-68 1 mile X Y Y 4 10

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Geometry 7-5 Angles of Elevation and Depression A. Definitions:

1. An angle of elevation is the angle where if you start horizontal and move upward.

2. An angle of depression is the angle where you start horizontal and move downward.

Ex 1: A man stands on a building and sees his friend on the ground. If the building is 70 m tall and the angle of depression is 35o, how far is the man from

the building?

Ex 2: A man notices the angle of elevation to the top of a tree is 60o, if he is 14 m

from the tree, how tall is the tree?

HW: Geometry 7-5 p. 374-376

8, 9, 11, 13, 14-18, 28-29, 31-35 odd, 36-39, 41-47 odd Hon: 19, 24

Angle of elevation

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Geometry 7-6 The Law of Sines

A. The Law of Sines - In trigonometry, the Law of Sines can be used to find missing parts of triangles that are not right triangles.

1. Let ABCV be any triangle with sides a, b, and c representing the measures of the sides opposite the angles with measures A, B and C respectively. Then sinA sinB sinC

a = b = c .

2. Proof of Law of Sines

Given: CD is an altitude of ABCV . Prove: sinA sinB

a = b

Statements Reasons

1.) CD is an altitude of ABCV 1.)_____________ 2.) ACDV and CBDV are rt V’s. 2.) Def of rt V’s. 3.) sinA h

b

= and sinB h a

= 3.) Def of sine 4.) b(sin )A = and h h a= (sin )B 4.) ______________ 5.) b(sin )A =a(sin )B 5.)_______________ 6.) sinA sinB

a = b 6.) Multiply each side by ____ Example 1: Find p. Round to the nearest tenth.

Example 2: Solve DEFV if m D∠ =112o, m F∠ =8o, and f = Round to 2

the nearest tenth.

C A B b c a C A B b h a D P R Q 17o 29o 8

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HW: Geometry 7-6 p. 381-383 17-35 odd, 38-39, 43, 46-58 Hon: 44-45

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Geometry 7-7 The Law of Cosines

A. The Law of Cosines - The Law of Cosines allows us to solve a triangle when the Law of Sines cannot be used.

1. Let ABCV be any triangle with sides a, b, and c representing the measures of the sides

opposite the angles with measures A, B and C respectively. Then the following equations are true: 2 2 2 2 cos a =b + −c bc A 2 2 2 2 cos b =a + −c ac B 2 2 2 2 cos c =a + −b ab C

2. You can use the Law of Cosines when you know two sides and the included angle.

3Example 1: Find c if b = 8, a = 6, and ∠ +C 48o

2 2 2 2 cos

c =a + −b ab C

3. You can use the Law of Cosines when you know all three sides and are looking for an angle.

Example 2: Use the Law of Cosines to solve for A∠ .

2 2 2 2 cos a =b + −c bc A HW: Geometry 7-7 p. 388-390 11-37 odd, 42, 46-47, 49-53 odd Hon: 39, 43, 57, 59 C A B b c a C A B 8 c 6 48o A B C 10 12 8

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