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

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

### 4

### 6

### y

### x

### z

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.

* _{a}*2

_{+}

*2*

_{b}_{= }

*2*

_{c}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 * _{a}*2

_{+}

*2*

_{b}_{= . }

*2*

_{c}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

### c

### b

### a

### 14

### 7

### x

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 _{60}o
x x
a

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: V*WXY* 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

**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

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

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

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 ABC*V 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 sin*A* sin*B* sin*C*

*a* = *b* = *c* .

2. Proof of Law of Sines

*Given: CD is an altitude of ABC*V .
Prove: sin*A* sin*B*

*a* = *b*

Statements Reasons

*1.) CD is an altitude of ABC*V 1.)_____________
*2.) ACD*V * and CBD*V are rt V’s. 2.) Def of rt V’s.
3.) sin*A* *h*

*b*

= and sin*B* *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.) sin*A* sin*B*

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

*Example 2: Solve DEF*V if *m D*∠ =112o_{, }_{m F}_{∠ =}_{8}o_{, 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 _{29}o
8

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

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