Transcript Chapter 7 - SchoolRack

A geometric sequence is found by multiplying
the previous number by a given factor, or
number.
5, 15, 45, 135,…
Set up a proportion to
compare the first 3
numbers
5
= 15
15
45
The cross
products are
equal!
The # in the
middle is the
GEOMETRIC
MEAN
I. Geometric Mean
• This is the geometric
mean:
a x

x b
• So
x  x  a b
x  a b
x  ab
2
The geometric mean has to be a positive number!
Example 1: Find the geometric
means for:
1 and 25
7 and 2
1
x

x
25
7
x 3
x


1
x
2 x
3
X² = 25
X=5
x² = 14
X = 14
3 and 1/3
x² = 1
X=1
REMEMBER THE PARTS OF
A RIGHT TRIANGLE?
II. Similar Triangles
Q
R
This is the
geometric mean!
S
P
QPS
QRP
PRS
III. Altitude Formula
• In a right triangle, the altitude is the
geometric mean of the two parts of the
hypotenuse
mean
h1
h2
Example 2: Find h.
9 h

h 25
h² = 225
9
25
h = 15
IV. Leg Formula
In a right triangle, the leg is the geometric mean of
the hypotenuse and the part of the hypotenuse
adjacent to that leg.
mean
h1
h2
Example 3: Find the value of a and b
6
a

a
4
4
a
2
a² = 24
A=2 6
b
6
b

b
2
b² = 12
B=2 3
V. The Pythagorean Theorem
• a 2 + b2 = c 2
• Pythagorean Triples: whole number side
lengths that fit the theorem.
Example 4:
6. Do 8,18, and 20 form a right
triangle?
7. Name two other Pythagorean
triples you can think of.
http://www.pisgah.us/organiz/geometry/accessoryinfo/Pyth-2.html
Try P 401: 7 - 14
7. 10
8. A. PTG PGA GTA
B. <PAG <TAG
9. X= 10 Y = 14
10. 213
11. 51
12. Yes
13. A. Yes 3 4 5
B. Each is a multiple of 3 4 5
C. Each is a multiple of 3 4 5
D. Yes: sides are multiples of the primitive triple
14. About 179.29 feet
7-3 Special Right Triangles
• I. Review
• What is the geometric mean of two
numbers a and b?
5
• Solve for x.
25
X
II.
RATIO:
The isosceles right triangle
( 45-45-90)
Looking for the hypotenuse?
Multiply the
leg
by √2
Looking for the leg?
Divide the
hypotenuse
by √2
Examples
• 1. Find AB and AC for isosceles triangle
ABC.
3
• 2. Find a and b.
a
7 2
b
• 3. Find a and b.
10
a
b
4. Find x and y.
19
x
y
III. The 30-60-90 right triangle
RATIO:
1:
3
:
2
You know the longest leg!
15
60°
DIVIDE BY
√3
DIVIDE BY
√3 AND
MULTIPLY
BY 2
You know the shortest leg!
30°
MULTIPLY
BY
MULTIPLY
BY 2
√3
18
You know the hypotenuse!
DIVIDE BY
2,
30°
40
MULTIPLY
BY √3
DIVIDE
BY 2
5. Find b and c.
30
c
8 3
60
b
You know the longer leg!
6. Find the indicated measures.
30
c
a
10
a
30
60
60
9
b
a=
a=
c=
b=
7. The measures of both legs of a
right triangle are 4. What is the
measure of the hypotenuse?
8. Find x.
CHALLENGE:
FIND THE AREA OF THE TRIANGLE!
• 9. The length of a diagonal of a square
is 20 centimeters. Find the length of a
side of a square
7-4 Special Ratios
I. NAMING SIDES IN A RIGHT TRIANGLE
II. Trig Ratios
A trigonometric ratio is a ratio of the lengths of two sides
of a right triangle.
A. THE SINE RATIO
B. THE COSINE RATIO
C. THE TANGENT RATIO
1. Compare the sine, the cosine, and the tangent
ratios for A in each triangle below.
8.5
B
17
A
8
15
A
7.5
B
4
C
C
SOLUTION
Large triangle
sin A =
cos A =
tan A =
Small triangle
opposite
hypotenuse
8
 0.4706
17
4
 0.4706
8.5
adjacent
hypotenuse
opposite
adjacent
15
 0.8824
17
8
 0.5333
15
7.5
 0.8824
8.5
4
 0.5333
7.5
Trigonometric
ratios are
frequently
expressed as
decimal
approximations.
2. Find the sine, the cosine, and the tangent of
the indicated angle.
S
R
5
SOLUTION
T
13
12
S
The length of the hypotenuse is 13. For S, the length of the opposite
side is 5, and the length of the adjacent side is 12.
opp.
5
=
hyp. 13
 0.3846
R
adj.
12
 0.9231
cos S =
=
hyp.
13
opp.5
sin
S
tan
S
=
opp.
5
 0.4167
=
=
adj.
12
T
13
12
adj.
hyp.
S
3. Find the sine, the cosine, and the
tangent of 45º.
SOLUTION
Because all such triangles are similar, you can make calculations simple by
choosing 1 as the length of each leg.
From the 45º-45º-90º Triangle Theorem, it follows that the length of the
hypotenuse is 2 .
sin 45º =
opp. =
hyp.
1
=
2
2
2
 0.7071
2
hyp.
1
cos 45º =
adj. =
hyp.
tan 45º =
opp.
1
=
adj.
1
1
=
2
=1
2
2
 0.7071
45º
1
4. Find the given length.
a.
b.
35 °
20
15
X
53 °
X
III. Finding the angle.
A. If you know the side lengths, and need to find
the angle, you just use the inverse button.
Tan X ° = opp
adj
= 6
6
20
X°
Press tan-1 (6 / 20)=
20
Your turn!
• 6. Find the angle.
• a.
b.
18
X°
14
32
X°
42
7-5 Angles of Elevation and
Depression
I. Angle of Elevation
Up from the point of reference - the
Horizon
Perspective to the Horizon
II. Angle of Depression
Down from the point of reference - the Horizon
1. FORESTRY You are measuring the height of a Sitka spruce tree in Alaska.
You stand 45 feet from the base of a tree. You measure the angle of elevation
from a point on the ground to the top of the tree to be 59°. To estimate the
height of the tree, you can write a trigonometric ratio that involves the height h
and the known length of 45 feet.
tan 59° =
opposite
adjacent
Write ratio.
tan 59° =
opposite
h
adjacent
45
Substitute.
45 tan 59° = h
Multiply each side by 45.
45(1.6643)  h
Use a calculator or table to find tan 59°.
74.9  h
Simplify.
The tree is about 75 feet tall.
2. ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in
Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels
on the escalator stairs, you can write a trigonometric ratio that involves the
hypotenuse and the known leg length of 76 feet.
sin 30° =
opposite
hypotenuse
Write ratio for sine of 30°.
sin 30° =
76opposite
hypotenuse
d
Substitute.
d sin 30° = 76
76
d=
sin 30°
76
d=
0.5
d = 152
Multiply each side by d.
d
76 ft
Divide each side by sin 30°.
30°
Substitute 0.5 for sin 30°.
Simplify.
A person travels 152 feet on the escalator stairs.
3. Find how high the
plane is from the ground.
16 km
12°
4. How far is the base of the
tower from the fire?
5°
43
ft
5. Find the angle of
elevation.
24 ft
11 ft