GEOMETRY JOURNAL CHAPTER 7 & 8

Christa Walters

9-5

May 12. 2011

RATIO•When using a ratio you express two numbers that are compared by division.• Can be written as: a to ba:bab

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PROPORTIONSA proportion is saying two ratios are equal

A ratio and a proportion are related because they need each other; why? Because when you state a proportion you use ratios.SOLVING PROPORTIONS:

1. You first cross multiply and then simplify by dividing the number across from the coefficient of x, that’s how you get your answer.

2. If the proportion involves an algebraic equation you first cross multiply, simplify by dividing, find the square roots of both sides and re write them as two equations and finally subtract two from both sides.

A CB D

Ex 1: Ex 2 : Ex3: Check: 5 = 45 6 = 3 x+2 = 24 10+2= 12Y 63 12 w 6 x+212(12)= 24(6)315= 45y 6(w)= 12(3) (x+2)^2 =144 144=144Y=7 w=6 x+2= +/- 12Check: check: x+2=12 pr x+2= -125(63)=7(45) 6(6)=12(3) x=10 or x= -14315=315 36=36

To make sure a proportion is correct, you have to plug the answer to the variable and solve the proportion by cross multiplying and if both sides equal the same number then it is correct

Similar polygons When two shapes are the same but have different sizes then they are two similar polygons. For them to be similar, their corresponding angles have to be congruent and also their corresponding sides need to be proportional. (∼)

Scale factorDescribes how much a figure can be reduced or enlarged. A ratio of two corresponding lengths in two figures.

examples 35: 10

35

10

50:20

50

20

60

30

60:30

Similar triangles with indirect measurement

When using similar triangles to find a measurement, the missing measurement you want to get to know one of the sides of the two triangles.

It’s an important skill to know how to use similar triangles while performing indirect measurements because you can use them in real life. For example you have to know the height of a tree you want to cut down so it doesn’t hit things near it. If you can’t climb all the way up to the tree what you can do is use indirect measurement. You first measure the shade of the tree. Then you measure your own height, and your own shade. You set it up as a proportion: tree shade/ your shade and tree height/ your own height. And just solve the proportion to get the missing measurement, the height of the tree.

Examples:

SCALE FACTOR. AREA AND PERIMETERTo find the AREA

you simplify the fraction of the two shapes. (big shape to small shape) and then square the whole fraction.

To find the PERIMETER you find the perimeter of each shape and then simplify the fraction .

:

Three trigonometric ratios.

•SIN: the ratio of opposite leg to hypotenuse

sinA: a/c SinB: b/c

•COSINE: the ratio of adjacent leg to hypotenuse

CosA: b/c CosB: a/c

•TANGENT: the ratio of opposite leg to adjacent leg

TanA: a/b TanB: b/a

You can use these three trigonometric ratios to find the angles and sides of right triangles

Examples 1 & 2: for each

Examples 3

How to remember them?

SOHCAHTOA

INE

PPOSITE

YPOTENUSE

OS

DJACENT

YPOTENUSE

AN

PPOSITE

DJACENT

What does it mean to solve a triangle?

Finding all the side and angle measurements of the triangle

ANGLES OF ELEVATION AND

DEPRESSIONAn angle of elevation is formed by a horizontal line from the point of sight to the point above that line. An angle of depression is formed by a horizontal line and a line of sight to a point that is under that line.People like air traffic controllers and boat controllers use angles of depression and elevation

X is the angle of depression

Y is the angle of elevation

y

Example 2:

Example 3:From bird to the ground= 2kmAngle 3= 32˚Angle 4= 320˚What distance does the bird have to fly to get to the person? Tan32= 2/xX= 2/tan32= 3.2

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