![Page 1: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/1.jpg)
Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all
![Page 2: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/2.jpg)
Chapter 9 Review
![Page 3: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/3.jpg)
Theorem 9.1: 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.
CNB~ANC~ACB:Then
CN altitude ACB; rt with ABC :Given
A
C
BN
![Page 4: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/4.jpg)
Theorem 9.2 (Geo mean altitude): When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.
CN altitude ACB; rt with ABC :Given
A
C
BN
AN CNCN BN
=
![Page 5: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/5.jpg)
Theorem 9.3 (Geo mean legs): 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.
CN altitude ACB; rt with ABC :Given
A
C
BN
AB ACAC AN
=
![Page 6: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/6.jpg)
Theorem 9.3 (Geo mean legs): 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.
CN altitude ACB; rt with ABC :Given
A
C
BN
AB ACAC AN
=AB BCBC BN
=
One way to help remember is thinking of it as a car and you draw the wheels.
Another way is hypotenuse to hypotenuse, leg to leg
![Page 7: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/7.jpg)
A
C
BN6 3
xy
w
z
6 + 3 = 9
w = 9
altGeo
x
x
x
x
23
18
3
6
2
legsGeo
y
y
y
y
63
54
6
9
2
legsGeo
z
z
z
z
33
27
3
9
2
![Page 8: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/8.jpg)
A
C
B
K
x
9
y z
w
15
16
259
x
x
legsGeo
z
z
z
z
20
400
16
25
2
altGeo
y
y
y
y
12
144
9
16
2
legsGeo
w
w
w
25
22599
15
15
![Page 9: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/9.jpg)
The Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
222 cba :Then
ACB rt with ABC :Given
a
c
b
![Page 10: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/10.jpg)
8 in
Find Area
![Page 11: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/11.jpg)
Converse of Pythagorean Theorem: If the square of the hypotenuse is equal to the sum of the squares of the legs, then the triangle is a right triangle.
ert triangl a is ABC :Then
cba with ABC :Given 222
a
c
b
B A
Cacute is ABC ;90CmThen
bac If 222
obtuse is ABC ;90CmThen
bac If 222
![Page 12: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/12.jpg)
12 6, 5, 2 ,1 ,3 9 8, 6, 8 11, 4,
neither)?(or obtuseor right, acute,it Is
16 64121 36 64 81 3 1 4 5 + 6 < 12
Neither
+ < + > + =
Obtuse Acute Right
Watch out, if the sides are not in order, or are on a picture, c is ALWAYS the longest side and should be by itself
![Page 13: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/13.jpg)
leg a as long as times2 is
hypotenuse the triangle,904545 aIn
904545
Theorem
legshort the times3 is leglonger
theand leg,short theas long as times2 is
hypotenuse the triangle,906030 aIn
906030
Theorem
45
45
x
x 2x
60
30
x2x
3xRemember, small side with small angle.
Common Sense: Small to big, you multiply (make bigger)
Big to small, you divide (make smaller)
For 30 – 60 – 90, find the smallest side first (Draw arrow to locate)
![Page 14: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/14.jpg)
Lots of examples
![Page 15: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/15.jpg)
sine sin
cosine cos
Tangent tan
These are trig ratios that describe the ratio between the side lengths given an angle.
ADJACENT
OP
PO
SIT
E
HYPOTENUSE
adjacent
OppositeA
Hypotenuse
adjacentA
Hypotenuse
OppositeA
tan
cos
sin
A
B
C
A device that helps is:
SOHCAHTOAin pp yp os dj yp an pp dj
![Page 16: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/16.jpg)
A
B
C14
539
B
B
B
A
A
A
tan
cos
sin
tan
cos
sin39
5
39
14
14
5
39
14
39
5
5
14
![Page 17: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/17.jpg)
x
y
20
3434sin
Find xHypotenuse
Look at what they want and what they give you, then use the correct trig ratio.
Opposite
opposite, hypotenuse
USE SIN!
hypotenuse
opposite x
20
Pg 845
Angle sin cos tan
34o .5592 .8290 .6745
Or use the calculator
205592.
x
x184.11
![Page 18: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/18.jpg)
x
y
20
3434cos
Find yHypotenuse
Look at what they want and what they give you, then use the correct trig ratio.
Adjacent
adjacent, hypotenuse
USE COS!
hypotenuse
adjacent y
20
Pg 845
Angle sin cos tan
34o .5592 .8290 .6745
Or use the calculator
208290.
y
y58.16
![Page 19: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/19.jpg)
4
30
x
Find x
Look at what they want and what they give you, then use the correct trig ratio.
AdjacentOpposite
Adjacent, Opposite, use TANGENT!
adjacent
oppositex tan
30
4
5.7tan x
Pg 845
Angle sin cos tan
81o .9877 .1564 6.3138 82o .9903 .1392 7.1154 83o .9925 .1219 8.1443
82x
If you use the calculator, you would put tan-1(7.5) and it will give you an angle back.
![Page 20: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/20.jpg)
Word Problems
• Hills, Buildings, Trees
![Page 21: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/21.jpg)
• Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all
![Page 22: Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all. Chapter 9 Review](https://reader035.vdocument.in/reader035/viewer/2022062409/5697bfd71a28abf838cae74f/html5/thumbnails/22.jpg)
• 14-23
• Geo mean legs, alt, pythag
• Pythag area of triangle
• 45-45-90, 30-60-90
• State trig ratios
• Trig word prob
• VECTORS!!