Geometry Journal Chapter 7 & 8

By: Jaime Rich

What is a ratio?

What is a proportion?

How are they related?

How to solve them and how to check if they are equal:

A comparison of two numbers by division.

An equation stating that two ratios are equal.

• You solve proportions by cross products or cross multiplication.• You check if they are equal if the product of the cross multiplication are equal to each other.

Proportions are related to ratios because proportions are made up of ratios.

Ratios

a:b

a to b

a/b

Proportions

5:10

5/10

5 to 10

a:b = c:d

a/b = c/d

a to b = c to d

5:10 = 1:2

5/10 = 1/2

5 to 10 = 1 to 2

Solving Proportions

a = c

b d

a = c

b d

ad = bc

Two polygons are similar when:

Scale Factor:

Their corresponding angles are congruent and their corresponding side lengths are proportional.

A similarity ratio is the ration of the lengths o f the corresponding sides of two similar polygons.

Shows how much a figure is enlarged or reduced.

168

36

9090 55 55

35

35

80

80

50

5050 50

6

4

12

18

Indirect Measurement:

It is an important skill because:

Any method that uses formulas, similar figures, and / or proportions to measure and object.

You can measure high, or big objects without physically measuring it with a ruler or Measuring tape. You can know distances from one place to another, and you can do many other things for planning for example to know where to build your house besides a tree pretending that tree may fall (you measure the height of the treewith indirect measurement and build the house away from it).

2 ft 6 ft

6 ft

h

2/6 = 6/h

2h = 36

H = 36/2

H = 18 ft.

4.5 ft

h

3 ft 30 ft

3/4.5 = 30/h

3h = 135

h = 135/3

h = 45

h

100 ft

50 ft2.5 ft

50/100 = 2.5/h

50h = 250

h = 250/50

h = 50 ft

How to use scale factor to find:Perimeter:

Area:

Theorem:

Perimeter of small figure

Perimeter of larger figure=

x

y

Area of small figure

Area of larger figure=

x

y----->Then square root x/y and square

that answer.

If the similarity ratio of two similar figures is a/b, then the ration of their perimeters is a/b, and the ratio of their areas is a/b squared.

PERIMETER

5

7

1015

21

3022 / 66 = 2 / 6 = 1/3

The scale is 1 to 3

3

2.51

4

1012

6.5/26 = 1/4

The scale is 1 to 4

1

1.54.5

3

2

3

6/9 = 1/1.5

The scale is1 to 1.5

Area

5

7

3

2.51

4

1012

1

1.54.5

3

2

3

21

3017.5/157.5 = √(1/9) = (1/3)squared

1.5/24 = √(1/16) = (1/4)squared

1/2.25 = √(1/2.25) = (1/1.5)squared

10

15

Three Trigonometric Ratios:Sine:

Cosine:

Tangent:

Ratio of the length of the leg opposite the angle to the length of the hypotenuse.

Ratio of the length of the leg adjacent to the angle to the length of the hypotenuse.

Ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle.

To solve a right triangle:

To solve a right triangle means:To find the measurements of sides and angles of right triangles.

You can use the Pythagorean theorem OR use sin, cos, or tan to solve it which is muchfaster and easier. You can also use the inverse trigonometric functions (sin-1, cos-1, and tan-1).

SIN / COS / TAN

R

5

T 12

13

S

SinR = 12/13 = 0.92

CosR = 5/13 = 0.38

TanR = 5/12 = 0.42

Cos(76) 0.2419218956

Ex 1

Ex 2

Ex 3Sin-1R = 67

Cos-1R = 68

Tan-1R = 23

Angle of Depression:

Angle of Elevation:

Angle formed by a horizontal line and a line of sight to a point below the line.

Angle formed by a horizontal line and a line of sight to a point below the line.

Angle of Depression

Angle of Elevation

< of elevation

< of elevation

< of depression

< of depression