9.1 – similar right triangles. theorem 9.1: if the altitude is drawn to the hypotenuse of a right...
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9.1 – Similar Right Triangles
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
BN
Set up Proportions
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
9.2 – Pythagorean Theorem
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
Given
Starfish both sides
Cross Multiplication (property of proportion)
Addition
Distributive Property =
Seg + post
Substituition prop =
• Pythagorean Triple is a set of three positive integers a, b, and c that satisfy the equation a2 + b2 = c2.
• Examples:– 3, 4, 5– 5, 12, 13– 7, 24, 25– 8, 15, 17– Multiples of those.
12
6
14
x
222 812 x
8
2
2
208
64144
x
x
x134
13
5
x
9
y
222 135 x
144
169252
2
x
x
12x
12
222 129 yy15
DON’T BE FOOLED, no right angle at top, can’t use theorems from before
8 in
Find Area
9.3 – The Converse of the Pythagorean Theorem
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
C
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
7 7, 7, 5,18 ,7 3 2, 1, 9 6, 5,
neither)?(or obtuseor right, acute,it Is
Reminders of the past. Properties of:Parallelograms Rectangles1) 1)2) 2)3) Rhombus4) 1)5) 2)6) 3)
Describe the shape, Why? Use complete sentences
24
725
9.4 – Special Right Triangles
• Rationalize practice
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
Find areas
9.5 – Trigonometric Ratios
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
C
BB
BA
AA
tancos
sintan
cossin
152
28
• Calculator CHECK– MODE!!!!!!!!!!! Should be in degrees– sin(30o) Test, should give you .5
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.
x20
50
68
x
x
1283
41
49
x
x20
506
8
x
y
y
40
70
x
34
17 70
x
1770cos
From the line of sight, if you look up, it’s called the ANGLE OF ELEVATION
From the line of sight, if you look down, it’s called the ANGLE OF DEPRESSION
ANGLE OF ELEVATIONANGLE OF DEPRESSION
For word problems, drawing a picture helps.
All problems pretty much involve trig in some way.
Mr. Kim’s eyes are about 5 feet two inches above the ground. The angle of elevation from his line of sight to the top of the building was 25o, and he was 20 feet away from the building. How tall is the building in feet?
25
feet20x
2025tan
x
326.9x
167.5
167.5 493.14
Mr. Kim is trying to sneak into a building. The searchlight is 15 feet off the ground with the beam nearest to the wall having an angle of depression of 80o. Mr. Kim has to crawl along the wall, but he is 2 feet wide. Can he make it through undetected?
80o
ft644.2
x
15)80tan(
Mr. Kim saw Mr. Knox across the stream. He then walked north 1200 feet and saw Mr. Knox again, with his line of sight and his path creating a 40 degree angle. How wide is the river to the nearest foot?
1200 ft
ft1007
1200)40tan(
x
The ideal angle of elevation for a roof for effectiveness and economy is 22 degrees. If the width of the house is 40 feet, and the roof forms an isosceles triangle on top, how tall should the roof be?
• DJ is at the top of a right triangular block of stone. The face of the stone is 50 paces long. The angle of depression from the top of the stone to the ground is 40 degrees (assume DJ’s eyes are at his feet). How tall is the triangular block?
9.6 – Solving Right Triangles
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.
Find x
Find all angles and sides, I check HW
Find all angles and sides
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