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

S

Q

R

U

V

T

W

A

CB

E

F

D2016

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 FACTOR

Examples1. 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 ft6 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

17

24

14

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

Ax

41⁰

Tan 41= 4000/xx= 4000/Tan 41x≈4601 ft

6.T

S Fx

7⁰

90 ft

Tan 7= 90/xx=90/Tan 7x≈ 733 ft