pg 586-587: 1 - 21 all; pg 582-584: 1 - 24 all. chapter 9 review
TRANSCRIPT
Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all
Chapter 9 Review
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
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
=
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
=
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
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
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
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
8 in
Find Area
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
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
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)
Lots of examples
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
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
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
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
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.
Word Problems
• Hills, Buildings, Trees
• Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all
• 14-23
• Geo mean legs, alt, pythag
• Pythag area of triangle
• 45-45-90, 30-60-90
• State trig ratios
• Trig word prob
• VECTORS!!