Ching Chi Tu

JOURNAL7 & 8

Proportion vs RatioRATIO – a ratio compares two numbers by

dividing. The ratio of two numbers can be written in various ways such as a to b, a:b, or a/b, where b doesn’t equal 0. For example the ratios of 3 to

4 can be represented as 3:4 or ¾

PROPORTION – a proportion is an equation stating that two ratios are equal. In this

proportion a/b = c/d, a and d are the extremes when b and c are the means. When a proportion is written as a:b = c:d, the extremes are in the first and last positions, which means that the

means are in the two middle positions.Proportions and ratios are related since proportions use ratios so inevitably, they both compare two number by division and can be used to create detailed miniature models.

How do you solvea proportion?

To solve a proportion, you need to use the Cross Products Property (in a proportion, if a/b = c/d and b and d don’t = 0, then ad = bc) so according to the Properties of Proportions, the proportion a/b = c/d is equivalent to the following:•ad = bc•b/a =d/c•a/c = b/d

EX1.3:4 = 12:x3x = 48x=16

EX2.5/17 = x/630 = 17xx = 1.76

EX3.297 to 2 = x to 1297 = 2xx = 148.5

Similar PolygonsTwo polygons are similar polygons iff

their corresponding angles are congruent and their corresponding side lengths are proportional. For them to be similar can

also mean that they have the same shape but not necessarily the same size.Scale Factor

- a scale factor is for describing how much a

figure is enlarged or deduced. For dilation, transformation that

changes size of a figure but not shape, with scale factor k, you can

find the image of a point by multiplying each coordinate by k: (a,b)

(ka, kb)

EX1EX1

90

90

90

90

90

90

90

90

4

6

2

3

EX2

157.5

15 7.5

189

21

EX2

EX3

EX3

15

45

3

9

75

25

INDIRECT MEASURES*Indirect measures are any method that uses formulas, similar figures, or proportions to measure an object.

To find indirect measures* with similar triangles, you have to follow some steps:1.Convert the measurements to a

single unit (if needed)2.Find the similar triangles3.Find the measurement you

need... (use cross products property)

This skill is important because if you were cutting down a tree near your house and you want to calculate if it is safe to do so, you may use this skill to find out the height of the tree.

4ft

EXA

MPL

E 1

3ft 20ft

h

¾ = 20/h3h = 80h = 26.667

5.3ft

EXA

MPL

E 2

4ft 29ft

h

4/5.3 = 29/h4h = 153.7h = 38.425

6ft

EXA

MPL

E 3

h 40ft

136ft

h/6 = 40/136136h = 240h = 1.765

Using Scale Factor to find

area / perimeterAREA

PERIMETER

(Area of small figure/Area of large figure)^2

Perimeter of small figure/Perimeter of large figure

EX1

4

3

12

9

2(4+3)/2(9+12)7/21=1/3

EX2

5

15

30

20

10

10

10+5+15/30+20+1030/60=1/2

EX3

EX3

Circumference 75

Circumference 37

75/37Already in simplest form…

Three Trigonometric

Ratios1. SINE – the sine of an angle is the ratio of the length of the leg opposite

the angle to the length of the hypotenuse.• sinA = opposite leg/hypotenuse = a/c• sinB = opposite leg/hypotenuse = b/c

2. COSINE – the cosine of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse.• cosA = adjacent leg/hypotenuse = b/c• cosB = adjacent leg/hypotenuse = a/c

3. TANGENT – the tangent of an angle is the ratio of the length of the leg opposite t he angle to the length of the leg adjacent to the angle.• tanA = opposite leg/adjacent leg = a/b• tanB = opposite leg/adjacent leg = b/a

You can also reverse it into (sin-1, cos-1, or tan-1)

S.O.H.C.A.H.T.O.A.

SINE.OPPOSITE.HYPOTENUSE.COSINE.ADJACENT.HYPOTENUSE.TANGENT.OPPOSITE.ADJACENT

A

B

C

ac

b

Since by the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with the same acute angle measure, and since a trigonometric ratio is a ratio of two sides of a right triangle, this can help us solve right triangles.

A

BC

6

5

4

EX1.SinB = 5/6EX2.CosB = 4/5EX3.TanB = 5/4

EX1.Sin-1B = 5/6 = 56.443EX2.Tan-1B = 5/4 = 51.34EX3.Cos-1B = 36.87

ANGLE OF ELEVATION/DEPRESSION

ELEVATION – the angle of elevation is the angle formed by a horizontal line and a line of sight to somewhere above the line. This is important when you are in a watch tower watching a plane descend in order to give correct directions.

DEPRESSION – the angle of depression is the angle formed by a horizontal line and a line of sight to somewhere below the line. This is important when you are a forest ranger and you need to stand high in an observation tower in order to see if a forest fire breaks out.

<of elevation

<of depression

<of depression

<of depression

<of elevation<of elevation