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JOURNAL 7 & 8. Maria Elisa Vanegas 9-5. RATIO. A ratio is a comparison of 2 things it could be 2 values. Examples A(-2,-1) B(4,3) rise 3-(-1) 4 2 run = 4-(-2) = 6 = 3. 3. A(-2,-2) B(2,2) rise -2-2 - 4 1 run = -2-2 = -4 = . 2. A(-1,3) B(1,4) rise 3-4 - 1 1 - PowerPoint PPT Presentation
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JOURNAL 7 & 8Maria Elisa Vanegas 9-5
RATIOA ratio is a comparison of 2 things it could be 2 values.Examples1. A(-2,-1) B(4,3) rise 3-(-1) 4 2 run = 4-(-2) = 6 = 3
2. A(-1,3) B(1,4) rise 3-4 -1 1 run = -1-1 = -2 = 2
3. A(-2,-2) B(2,2) rise -2-2 -4 1 run = -2-2 = -4 =
PROPORTIONA proportion is simply a equation that tells us that 2 ratios are equal to each other. You solve proportions by cross multiplying the given fractions and then simplifying. You can check by inserting the variable to the equation and verifying.Examples1. 5 45 y = 63 5(63)=y(45) 315=45y y=7
2. x+2 2 4 6 = x+2 (x+2)²=6(24) (x+2)²=144 x+2= +/- 12 x+2=+/- 12 x= 10 or -14
3. 16 x-1 x-1 = 4 16(4)=x²-2 64=x²-2 ∫66=∫x² ∫66=x
These 2 are related because they both involve ratios.
SIMILAR POLYGONSPolygons are similar iff they have corresponding angles that are congruent and their corresponding side lengths are proportional.
Examples 1-3Determine weather the polygons are similar. If so, write the similarity ratio and a similarity statement.1.
2.
<P congruent <T, <Q congruent <U ,<R congruent <V, <S congruent <W
PQ = 12 = 3 PS = 4 = 2TU 16 4 TW 6 3
124
166
P
SQ
R
U
V
T
W
A
CB
E
FD2016
24
1812
15
AB = 20 = 4 BC = 24 = 4 AC = 16 = 4DE 15 3 EF 18 3 DF 12 3
<A congruent <D, <B congruent <E ,<C congruent <F
3. EH = 30 = 2 EF = 90 = 2 AD 45 3 AB 135 3
135
45
90
30
A B
D CE F
H G
The only thing that these does is that it helps determine how much something is enlarged or reduced.
SCALE FACTORExamples1. Multiply the vertices of the photo A B C D by 3/2.
B (0,4)
A (0,0)
C (3,4)
D (3,0)
A(0,0)A(0 [3/2], 0[3/2])A(0,0)B(0,4)B(0[3/2], 4[3/2])B(0,6)C(3,4)C(3[3/2], 4[3/2])C(4.5,6)D(3,0)C(3[3/2],0[3/2])D(4.5,0)
A(0,0)
B(0,6) C(4.5,6)
D(4.5,0)
2.
A(0,0)A(0[1/2], 0[1/2])A(0,0)B(0,6)A(0[1/2], 6[1/2])B(0,3)C(4.5,6)C(4.5[1/2], 6[1/2])C(2.25,3)D(4.5,0)D(4.5[1/2], 0[1/2])D(2.25,0)
A(0,0)
B(0,6) C(4.5,6)
D(4.5,0)
A(0,0)
B(0,3) C(2.25,3)
D(2.25,0)
Multiply the vertices of the photo A B C D by 1/2.
3. Multiply the vertices of the photo A B C D by 4/3. ROUND IF NEEDED
A(0,0)A(0[4/3], 0[4/3])A(0,0)B(0,8)B(0[4/3], 8[4/3])B(0,11)C(4,8)C(4[4/3], 8[4/3])C(5.3,11)D(4,0)D(4[4/3], 0[4/3])D(5.3,0)
A(0,0)
A(0,0)
B(0,8) C(4,8)
D(4,0)
B(0,11) C(5.3,11)
D(5.3,0)
o Right Triangle Similarity if you draw an altitude from the vertex of the right angle of a right triangle, you form 3 similar right triangles.
o You do this by using ratios like shortest side/longest side of 2 similar triangles then you simplify.
o This is an important skill because if someday you want to cut a tree of your house you have got to know how long it is so it doesn't crushes you house.x
y
8
z
3
Examples Find all of the sides
1. x = 3 1.125 = y 3.2 = 3 3 8 y 9.125 1.125 z 8x=9 ∫y² = ∫10.27 3.375 = 3.2z x= 1.125 y=3.2 3.2 3.2 z=1.1
Indirect Measurement
6 ft 6 ft
30 ft
3. Find the height of the tower.
6 = 30 30= x6x = 9006 6X = 150 150 + 6 = Height= 156 ft
x
2. Find the height of the Ceiba.
8 ft 8 ft
x
45 ft
8 = 4545= x2025= 8x8 8253.125= x253.125+8= height =261.125 ft
Perimeter and Area
o Area- first you have to simplify the fraction of both shapes after you have done that you square the fraction.
o Perimeter- first you find the perimeter of each shape with that you create a fraction of each perimeters then simplify.
6 4 312
1 7
2414
6(4)=24 16 = 24(4)=16 24 3
3(2)+12(2)=301(2)+7(2)=1616 8 30 = 15
14(4)= 5624(24)=9656= 796 12
1.Sides40&2540/25 = 8/5(8/5)² = 64/25
2.Sides 30&1230/12=5/2(5/2) ²= 25/4
3.Sides 94&8694/86=47/43(47/43) ²= 2209/1849
TRIGONOMETRIC RATIOSo Trigonometric= the study of triangleso Sin A= Opposite/Hypotenuseo Cos A= Adjacent/Hypotenuseo Tan A= Opposite/Adjacento Solving a triangle means finding all of the angles and all of the
sides.o These are useful to solve a right triangle because it helps you find
the angles and the sides .Examples
Write the ratio as a # and decimal rounded.R
ST
13
12
5Sin R= 12/13 ≈ 0.92Cos T= 5/13 ≈ 0.38Tan S= 5/12 ≈ 0.42
100 m40⁰
Tan 40 = x__ 100100 (Tan 40) = x83.90
x
B
42⁰
x12
Sin 42 = x/1212(Sin 42)= x = 8.02
CA
B
24
257
Cos A= 24/25 ≈ 0.96Tan B= 24/7 ≈ 3.42Sin B= 24725 ≈ 0.96
x
y
z12.6 cm38⁰
Cos 38= 12.6/YZYZ= 12.6/Cos 38YZ= 15.99 cm
ANGLE OF ELEVATION & ANGLE OF DEPRESSION
o Angle of Elevation is a straight line going horizontally and another line that’s ABOVE the horizontal pointing somewhere, which together form the angle.
o Angle of Depression is a straight line going horizontally and another line that’s BELLOW the horizontal pointing somewhere, which together form the angle.
Angle of Depression
Angle of Elevation
Clasify each angle as angle of depression or elevation
ball<1
<2<3
<41. <1 is angle of
elevation2. <2 is anlge of
depression3. <3 is angle of
elevation4. <4 is angle of
depression
5.P
A x41⁰
Tan 41= 4000/xx= 4000/Tan 41x≈4601 ft
6. T
S Fx
7⁰
90 ftTan 7= 90/xx=90/Tan 7x≈ 733 ft