Download - Lesson 9.1 Using Similar Right Triangles
![Page 1: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/1.jpg)
![Page 2: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/2.jpg)
Lesson 9.1Using Similar Right Triangles
Students need scissors, rulers, and note cards.
Today, we are going to……use geometric mean to solve problems involving similar right triangles formed by the altitude drawn to the hypotenuse of a right triangle
![Page 3: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/3.jpg)
With a straight edge, draw one diagonal of the note card.
Draw an altitude from one vertex of the note card to the diagonal.
Cut the note card into three triangles by cutting along the segments.
![Page 4: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/4.jpg)
C
BAA D
B
DB
Chy
pote
nuse
hyp
hyp
short leg
short legshortlo
ng
leg
lon
g le
g
lon
g
Color code all 3 sides of all 3 triangles on the front and back.
![Page 5: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/5.jpg)
Arrange the small and
medium triangles on top of the
large triangle like this.?
![Page 6: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/6.jpg)
If the altitude is drawn to the hypotenuse of a right
triangle, then…
Theorems 9.1 – 9.3
![Page 7: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/7.jpg)
Theorem 9.1
…the three triangles formed are similar to each other.
![Page 8: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/8.jpg)
BDBDCD
AD=
BD is a side of the medium and a side of the small
A
B
C D
![Page 9: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/9.jpg)
D AC
B
x
m n
xx=mn
BDBD=
CDAD
![Page 10: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/10.jpg)
Theorem 9.2
…the altitude is the geometric mean of the two
segments of the hypotenuse.
D AC
B
x
m n
xx=mn
![Page 11: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/11.jpg)
____ is the geometric mean of ____ and ____
When you do these problems, always tell yourself…
![Page 12: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/12.jpg)
D AC
B
x
8 3
xx=
83
1. Find x.
x ≈ 4.9
![Page 13: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/13.jpg)
D AC
B
4
8 x
44=
8x
2. Find x.
x = 2
![Page 14: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/14.jpg)
CBCD CB
CA=
A
B
C D
CB is a side of the large and
a side of the medium
![Page 15: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/15.jpg)
D AC
B
x
m
h
xx=mh
CBCB=
CDCA
![Page 16: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/16.jpg)
ABAD AB
AC=
A
B
C D
AB is a side of the large and
a side of the small
![Page 17: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/17.jpg)
D AC
B
x
nh
xx=nh
ABAB=
ADAC
![Page 18: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/18.jpg)
Theorem 9.3
…the leg of the large triangle is the geometric mean of the
“adjacent leg” and the hypotenuse.
D AC
B
x
m n
xx=mh
y
h
yy=nh
![Page 19: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/19.jpg)
____ is the geometric mean of ____ and ____
When you do these problems, always tell yourself…
![Page 20: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/20.jpg)
D AC
B
x
9
14
xx=9
143. Find x.
x ≈ 11.2
![Page 21: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/21.jpg)
D AC
B
x
410
xx=4
104. Find x.
x ≈ 6.3
![Page 22: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/22.jpg)
5. Find x, y, zx y z
9 4
xx=9
13
x ≈ 10.8
yy=94
y = 6
zz=4
13
z ≈ 7.2
![Page 23: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/23.jpg)
33=x56. Find h.
x = 1.8
h4
3
5
x1.8
3.2h
h=
3.2
1.8
h = 2.4
![Page 24: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/24.jpg)
![Page 25: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/25.jpg)
Lesson 9.2 & 9.3 The Pythagorean Theorem
& Converse
Today, we are going to……prove the Pythagorean Theorem…use the Pythagorean Theorem and
its Converse to solve problems
![Page 26: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/26.jpg)
Theorem 9.4Pythagorean Theorem
hyp2 = leg2 + leg2 (c2 = a2 + b2 )
![Page 27: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/27.jpg)
2. Find x.
8
20
10 x
y
102 = y2 + 82
100 = y2 + 6436 = y2
14
x2 = 142 + 82
x2 = 196 + 64
x2 = 260
x ≈ 16.1
y = 6
Why can’t we use a geo mean proportion?
![Page 28: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/28.jpg)
A Pythagorean Triple is a set of three positive
integers that satisfy the equation c 2 = a 2 + b 2.
The integers 3, 4, and 5 form a Pythagorean Triple because
52 = 32 + 42.
![Page 29: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/29.jpg)
Theorem 9.5Converse of the
Pythagorean Theorem
If c2 = a2 + b2, then the triangle is a right triangle.
![Page 30: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/30.jpg)
The “hypotenuse” is too short for the opposite angle to be 90˚ c2 < a2 + b2
The “hypotenuse” is too long for the opposite angle to be 90˚
c2 > a2 + b2
The hypotenuse is the perfect length for the opposite angle to be 90˚
c2 = a2 + b2
![Page 31: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/31.jpg)
Theorem 9.6
If c2 < a2 + b2, then the triangle is an acute
triangle.
![Page 32: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/32.jpg)
Theorem 9.7
If c2 > a2 + b2, then the triangle is an obtuse
triangle.
![Page 33: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/33.jpg)
How do we know if 3 lengths can represent the side lengths
of a triangle?
L < M + S
![Page 34: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/34.jpg)
What kind of triangle?
L2 = M2 + S2
L2 < M2 + S2
L2 > M2 + S2
Right
Acute
Obtuse
Can a triangle be formed?
L = M + S
L < M + S
L > M + S
No can be formed
Yes, can be formed
No can be formed
![Page 35: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/35.jpg)
Do the lengths represent the lengths of a triangle? Is it a right triangle, acute triangle, or obtuse triangle?
3. 10, 24, 26
4. 3, 5, 7
5. 5, 8, 9
262 = 102 + 242
72 > 32 + 52
92 < 52 + 82
right triangle
obtuse triangle
acute triangle
26 < 10 + 24?
7 < 5 + 3?
9 < 8 + 5?
![Page 36: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/36.jpg)
92 = 52 + 56 2
Do the lengths represent the lengths of a triangle? Is it a right triangle, acute triangle, or obtuse triangle?
6. 5, 56 , 9
7. 23, 44, 70
8. 12, 80, 87 872 > 122 + 802
right triangle
not a triangle
obtuse triangle
70 < 23 + 44?
87 < 12 + 80?
9 < 5 + 56 ?
![Page 37: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/37.jpg)
Find the area of the triangle.
9.
6
9
81 = x2 + 36
92 = x2 + 62
45 = x2
6.7 = xA ≈ ½ (6)(6.7)
≈ 20.12 units2
![Page 38: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/38.jpg)
Find the area of the triangle.
10.13
20400 = x2 + 169
202 = x2 + 132
231 = x2
15.2 = xA ≈ ½ (13)(15.2)
≈ 98.8 units2
![Page 39: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/39.jpg)
Find the area of the triangle.
11.
49 = x2 + 4
72 = x2 + 22
45 = x2
6.7 = xA ≈ ½ (4)(6.7)
≈ 13.4 units2
7
2
x
![Page 40: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/40.jpg)
How much ribbon is needed using Method 1?
Which method requires less ribbon?
The diagram shows the ribbon for Method 2. How much ribbon is needed to wrap the box?
(3+12+3+12) + (3+6+3+6) = 48 in.
?
?
30
18
35 in
302 + 182 =
![Page 41: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/41.jpg)
Project ideas…A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. How far up on the building will the ladder reach?
A 50-meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters from the base. How long is the cable?
The library is 5 miles north of the bank. Your house is 7 miles west of the bank. Find the distance from your house to the library.
![Page 42: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/42.jpg)
Project ideas…Playing baseball, the catcher must throw a ball to 2nd base so that the 2nd base player can tag the runner out. If there are 90 feet between home plate and 1st base and between 1st and 2nd bases, how far must the catcher throw the ball?
While flying a kite, you use 100 feet of string. You are standing 60 feet from the point on the ground directly below the kite. Find the height of the kite.
![Page 43: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/43.jpg)
![Page 44: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/44.jpg)
Lesson 9.4Special Right Triangles
Today, we are going to……find the side lengths of special right triangles
![Page 45: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/45.jpg)
45˚
45˚
5
xy
1. Find x. Use the Pythagorean Theorem to find y. Leave y in simplest radical form.
x = 5y 5 2
![Page 46: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/46.jpg)
45˚
45˚
9
xy
2. Find x. Use the Pythagorean Theorem to find y. Leave y in simplest radical form.
x = 9y 9 2
![Page 47: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/47.jpg)
Do you notice a pattern?
45˚
45˚
5
55
45˚
45˚
9
992 2
![Page 48: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/48.jpg)
In a 45˚- 45˚- 90˚ Triangle,
hypotenuse = leg 2
Theorem 9.845˚- 45˚- 90˚
Triangle Theorem
45˚
45˚
x
xx 2
![Page 49: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/49.jpg)
45˚
45˚
7
xy
3. Find x and y.
x = 7y = 7 2
![Page 50: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/50.jpg)
45˚
45˚
y
x3
4. Find x and y.
2x = 3
y = 3
![Page 51: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/51.jpg)
45˚
45˚
y
x10
5. Find x and y.
x = 5 2
x =
2
10
2
y = 5 2
![Page 52: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/52.jpg)
6. Label the measures of all angles.Find x. Use the Pythagorean Theorem to find y in simplest radical form. 66
6
x
y
x = 3 60˚60˚
30˚30˚
y = 3 3
![Page 53: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/53.jpg)
7. Label the measures of all angles.Find x. Use the Pythagorean Theorem to find y in simplest radical form. 88
8
x
y
x = 4 60˚60˚
30˚30˚
y = 4 3
![Page 54: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/54.jpg)
Do you notice a pattern?
30˚
60˚
3
63 3
60˚
30˚
8
4
34
![Page 55: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/55.jpg)
hypotenuse = 2 short leg
long leg = short leg 3
Theorem 9.930˚-60˚-90˚
Triangle Theorem
60˚
30˚
2s
s
3s
In a 30˚-60˚-90˚ Triangle,
![Page 56: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/56.jpg)
8. Find x and y.
10
x
y
60˚
30˚x = 5
y = 5 3
![Page 57: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/57.jpg)
9. Find x and y.
x
10
y
60˚
30˚x = 20
y = 10 3
![Page 58: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/58.jpg)
10. Find x and y.
y
x
12
60˚
30˚
3
x = 12
y = 24
![Page 59: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/59.jpg)
11. Find x and y.
x
y
12
60˚
30˚
y 12
3
y = 4 3
x = 8 3
![Page 60: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/60.jpg)
12. In regular hexagon ABCDEF, find x and y.
36012
y =
y = 30
x = 60
![Page 61: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/61.jpg)
13. Find x and y.
x = 24
y = 12 3
![Page 62: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/62.jpg)
14. Find x and y.
x = 8
y = 8 2
![Page 63: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/63.jpg)
![Page 64: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/64.jpg)
Lesson 9.5 & 9.6Trigonometric
RatiosToday, we are going to……find the sine, cosine, and tangent of an acute angle…use trigonometric ratios to solve problems
![Page 65: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/65.jpg)
Trigonometric Ratios
sinecosinetangent
![Page 66: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/66.jpg)
A
B
C
hypotenuse
adjacent to A
opposite A
____ is the hypotenuse____ is opposite A____ is adjacent to AABBCAC
![Page 67: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/67.jpg)
A
B
C
hypotenuse
opposite B
adjacent to B
____ is the hypotenuse____ is opposite B____ is adjacent to BABACBC
![Page 68: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/68.jpg)
sin A =
leg opposite A
hypotenuse
![Page 69: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/69.jpg)
cos A =
leg adjacent to A
hypotenuse
![Page 70: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/70.jpg)
tan A =
leg opposite A
leg adjacent to A
![Page 71: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/71.jpg)
SOHCAHTOA
Sine is Opp / Hyp
Cosine is Adj / Hyp
Tangent is Opp / Adj
![Page 72: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/72.jpg)
1.
sin A = = 0.38465
13
12
513
A
Opp
Hyp
Adj
cos A =
tan A = = 0.4167 5 12
= 0.9231 12 13
B
![Page 73: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/73.jpg)
2.
sin B = = 0.923112 13
12
513
A
Adj
Hyp
Opp
cos B =
tan B = = 2.400012 5
= 0.3846 5 13
B
![Page 74: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/74.jpg)
Find the Sine, Cosine, and Tangent.
3. sin 32° =
5. tan 32° =
0.5299
0.5299
0.6249
4. cos 58°=
![Page 75: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/75.jpg)
6. Find x and y to the nearest tenth.
OPP
ADJ
HYPsin 42˚ =
cos 42˚ = x121
12
x
y42°
x = 12 cos 42˚
x ≈ 8.9
y121
y = 12 sin 42˚ y ≈ 8.0
![Page 76: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/76.jpg)
7. How tall is the tree?
64.34 ft
tan 65˚ = x301
x = 30 tan 65˚
65°30 ft
xOpp
Hyp
Adj
![Page 77: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/77.jpg)
8. Find x to the nearest tenth.
42° 12
x
x = 13.3
tan 42˚ = 12 x1
12 = x tan 42˚
x =tan 42˚
12 ? 48˚
tan 48˚ = x121
x = 12 tan 48˚
x = 13.3
Opp
Hyp
AdjAdj
Hyp
Opp
![Page 78: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/78.jpg)
Use Inverse Sine, Inverse Cosine, and Inverse Tangent.
9. sin A =1625
10. cos A =4553
11. tan A = 0.4402
m A =
m A =
m A =
40°
32°
24°
![Page 79: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/79.jpg)
Use Inverse Sine, Inverse Cosine, and Inverse Tangent.
12. sin-1(0.7660)= A
14. tan-1(11.4300) = A
m A =
m A =
m A =
13. cos-1 513
= A
50°
67°
85°
![Page 80: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/80.jpg)
Calculator language:
cos-1 513
= A
cos A = 513
Human language:
![Page 81: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/81.jpg)
15. Solve the right triangle.
30
16A
C
B
mA = 28°
mB = 62°
AB = 34
tan A = 1630
(AB)2 = 302 + 162
opp
adj
![Page 82: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/82.jpg)
16. Solve the right triangle.
10
6A
C
B
mA = 37°
mB = 53°
AC = 8
sin A = 610
102 = (AC)2 + 62
opp
hyp
![Page 83: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/83.jpg)
angle of elevation
angle of depression
A
B
A
B
C
C
![Page 84: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/84.jpg)
?
20°
tan 20°=x8
x = 8 tan 20°
x = 2.9 ft
![Page 85: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/85.jpg)
sin 45°= 3026+x
sin 45° sin 45° 30 = (26+x) sin 45°
42.43 = 26+x
16.43 = x
For the most comfortable height, the handle should be 16.43 inches.
![Page 86: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/86.jpg)
![Page 87: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/87.jpg)
![Page 88: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/88.jpg)
12 miles
44˚
x
sin 28° = 60 x
x 127.8 cos 22° =
300 x
x 323.56
x 16.68
tan 44° = x 12
x 11.59
12 miles
44˚
x
cos 44° = 12 x
![Page 89: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/89.jpg)
x 116.6
tan 25° = x 250
x 31.2
tan 58° = 50 x
x 84.8
cos 32° = x 100
x 46.7
sin 40° = 30 x
![Page 90: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/90.jpg)
Lesson 9.7 Vectors
Today, we are going to……find the magnitude and the
direction of a vector …add two vectors
![Page 91: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/91.jpg)
The magnitude of a vector PQ is the distance from the
initial point to the terminal point and is written | PQ |.
?
In other words, it is “the length of the vector”We use absolute value
symbols because length cannot be negative.
![Page 92: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/92.jpg)
To find the magnitude of a vector…
Step 1: Identify the vector
component form X, Y
Step 2: Find the magnitude by
simplifying (X)2 + (Y)2
![Page 93: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/93.jpg)
The component form of AB is
__________-5, -3
For example, to make AB, we go left 5 units and down 3 units.
To find the magnitude of AB,
we simplify (-5)2 + (-3)2
and get 34 5.8
![Page 94: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/94.jpg)
+4
+5?
A
B
| AB | ≈ 6.4
1. Find the magnitude of the vector.
4,5
(4)2 + (5)2
First, write the component form of
the vector.
Now, find the magnitude using the
formula.
![Page 95: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/95.jpg)
-7
+5
| AB | ≈ 8.6
A
B2. Find the magnitude of the vector.
-7,5(-7)2 + (5)2
magnitude formula?component form?
![Page 96: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/96.jpg)
The direction of a vector is determined by the angle it
makes with a horizontal line.
45˚
45˚ southeast
![Page 97: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/97.jpg)
30˚ southwest30˚
Identify the direction of the vector.
3.
4.
50˚ northwest
50˚
![Page 98: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/98.jpg)
+5?
5. Find the direction of the vector
51° northeast+4
tan A = 54
m A = 51˚
opposite
adjacent
hypo
tenu
se
mark the hypotenuse
mark the opposite leg mark the adjacent leg
sin, cos, or tan?
![Page 99: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/99.jpg)
+5
6. Find the direction of the vector
36° northwest
?
-7
tan A =
m A = 36˚
5 7
oppo
site
adjacent
hypotenuse
hypotenuse?opposite leg?
adjacent leg?
sin, cos, or tan?
Use +7 because it is length
![Page 100: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/100.jpg)
Step 3: Direction
Step 2: Magnitude
7. Find the magnitude and direction
of AB if A (1,2) and B (4, 6).We can do this without drawing the vector!= X , Yx2 – x1 , y2 – y1
Step 1: Component Form4 – 1 , 6 – 2 = 3 , 4
(X)2 + (Y)2| AB | =
| AB | = (3)2 + (4)2 25 5=tan A =
|Y| |X|
53 northeast
tan A =
4
3
![Page 101: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/101.jpg)
LOOK at the component form!
How do we know if the vector is north or south, east or west
without sketching it?
+,+ is right and up northeast-,+ is left and up northwest
-,- is left and down southwest+,- is right and down southeast
![Page 102: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/102.jpg)
+,+ right,up
northeast
-,+ left ,up
northwest
left ,down-,-
southwest
right,down +,-
southeast
It might help to think about a map of the US!
![Page 103: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/103.jpg)
8. Find the magnitude and direction
of AB if A (-3,3) and B (4, -5).= X , Y
(X)2 + (Y)2
tan A =
|Y| |X|
49 southeast
x2 – x1 , y2 – y1
| AB | =
= 7 , - 84 – –3 , – 5 – 3(7)2 + (-8)2=
| AB | = 113 10.6
tan A =
87
![Page 104: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/104.jpg)
Component Form= X , Y
Magnitude(X)2 + (Y)2
Direction
tan A =
|Y| |X|
A north/south - east/west
x2 – x1 , y2 – y1
| AB | =
![Page 105: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/105.jpg)
Adding Vectors in Component Form
![Page 106: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/106.jpg)
2,-3 + 5,4 =
5,42,-3
7,1
7,19.
Pick any starting pointmove right 2 and down 3from that point, go right 5 and up 4
instead of taking this route, you could take a short cut
See a pattern?
![Page 107: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/107.jpg)
9,-2
-3,5 6,3
Pick a starting pointmove left 3 and up 5from that point, go right 9 and down 2
short cut?
See a pattern?-3,5 + 9,-2 = 6,310.
![Page 108: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/108.jpg)
12.
11.
3, 4 , 6,5 , 2,1u v w ------------- -
u v
u w
------------- - 9 , 1
1 , - 3
![Page 109: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/109.jpg)
12 miles
44˚
x
sin 28° = 60 x
x 127.8 cos 22° =
300 x
x 323.56
x 16.68
tan 44° = x 12
x 11.59
12 miles
44˚
x
cos 44° = 12 x
![Page 110: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/110.jpg)
x 116.6
tan 25° = x 250
x 31.2
tan 58° = 50 x
x 84.8
cos 32° = x 100
x 46.7
sin 40° = 30 x
![Page 111: Lesson 9.1 Using Similar Right Triangles](https://reader035.vdocuments.net/reader035/viewer/2022062304/56813ab8550346895da2bd81/html5/thumbnails/111.jpg)