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

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
a x
positive number x where: = .
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.
A
D
VABC : VA ___ : VB ___
C
B
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.
A
VADB : VBDC
w
D
AD BD
=
DB DC
x
a
B
C
w a
=
a x
Ex 3: Find the length of the altitude, if the following is true.
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.
A
w
VABC : VADB : VBDC
D
AD AB
=
b
x
AB AC
a
C
B
w
b
=
b x + w ( hyp)
Ex 4: Find the length of the given sides if the following is true.
4
x
y
6
z
HW: Geometry 7-1 p. 346-348
13-32 all, 35-38 all, 42-43, 49-50, 55-65 odd
Hon: 34, 44,
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.
c
a
a 2 + b2 = c 2
b
Ex 1: Find x.
14
7
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 + b 2 = c 2 .
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
Geometry 7-3 Special Right Triangles
A. 45o − 45o − 90o Triangles
d
x
Do the Pythagorean Theorem (solve for d)
a 2 + b2 = c 2
x2 + x2 = d 2
x
1. Theorem 7-6 - In a 45o − 45o − 90o triangle,
the length of the hypotenuse is 2 times as
long as a leg.
s 2
s
leg 2 = hypotenuse
s
Example 1: Find the length of the sides of the triangle.
6
Example 2: If the leg of a 45o − 45o − 90o triangle is 12 units, find the length
of the hypotenuse.
B. 30o − 60o − 90o Triangles
1. What is the relationship between the short leg
of a 30o − 60o − 90o , triangle and the hypotenuse?
C
short leg (___) = hypotenuse
2. Let’s do Pythagorean Theorem to solve for a.
a 2 + x 2 = (2 x)2
short leg (____) = long leg
a
A
60o
x
D
x
60o
B
1. Theorem 7-7 - In a 30o − 60o − 90o 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.
30o
2n
n 3
n
Example 3: Find AB and AC.
B
60o
A
Example 4: VWXY is a
30o − 60o − 90o 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
12
C
60o
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
B
sin A
leg opposite ∠A a
=
hypotenuse
c
cos A
leg adjacent to ∠A b
c
a
=
hypotenuse
c
tan A
leg opposite to ∠A a
=
C
adjacent
b
A
b
Example 1: Find the sin S, cos S, tan S, sin E, cos E, tan E.
M
6
8
S
10
Ex 2: Solve the triangle
A
6
B
35o
C
E
Ex. 3: Solve the triangle
X
10
Y
4
Y
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?
1 mile
HW: Geometry 7-4 p. 368-370
18-48, 63-64, 69-81 odd
Hon: 55-58, 65-68
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.
Angle of elevation
2.
An angle of depression is the angle where you start horizontal and move
downward.
Angle of depression
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
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.
C
1. Let VABC 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.
sin A sin B sin C
=
=
Then
.
a
b
c
b
A
B
c
C
2. Proof of Law of Sines
b
Given: CD is an altitude of VABC .
sin A sin B
=
Prove:
a
b
Statements
1.) CD is an altitude of VABC
2.) VACD and VCBD are rt V’s.
h
h
3.) sin A = and sin B =
b
a
4.) b(sin A) = h and h = a (sin B )
5.) b(sin A) = a (sin B )
sin A sin B
=
6.)
a
b
a
a
h
A
B
D
Reasons
1.)_____________
2.) Def of rt V’s.
3.) Def of sine
4.) ______________
5.)_______________
6.) Multiply each side by ____
Q
Example 1: Find p. Round to the nearest tenth.
8
P
17o
29o
R
Example 2: Solve VDEF if m∠D = 112o , m∠F = 8o , and f = 2 Round to
the nearest tenth.
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.
C
1. Let VABC 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:
a 2 = b 2 + c 2 − 2bc cos A
b 2 = a 2 + c 2 − 2ac cos B
c 2 = a 2 + b 2 − 2ab cos C
b
A
a
B
c
2. You can use the Law of Cosines when you know two sides and the
included angle.
C
3Example 1: Find c if b = 8, a = 6, and ∠C + 48
o
c 2 = a 2 + b 2 − 2ab cos C
48o
8
A
6
B
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 .
a 2 = b 2 + c 2 − 2bc cos A
A
8
B
HW: Geometry 7-7 p. 388-390
11-37 odd, 42, 46-47, 49-53 odd
Hon: 39, 43, 57, 59
10
12
C