Chapter 7 and 8

By: Ou Suk Kwon

Ration• Comparing 2 numbers that are written: • A to B • A / B• A:B

Examples of ratio

The ratio of dirty cars to clean cars is 1:2

The ratio of black guns to white gun is 6:1

The ratio of red fruits to another color fruits are 4:8

Proportions • Proportion is an equation that states that two

ratios are equal to each other. (compares ratio stating that they are same.)

Examples of proportions

Proportion 9:2=18:4

Proportion4:6=2:3

examples

Proportion 1:2=3:6

how to solve proportionsIf there are 2 variables

Cross multiply the fraction then, square root both sidesUse + and – root to solve for xThe answer for X will be 2 numbers

If there is only one variableCross multiply the fraction and then divide.

Examples1) 16/2=X/4First cross multiply so the answer will be 2x=64Then divide which is X=32

2) 10/x+2=x+2/2So here, cross multiply: (x+2)2=20Then multiply which is x+2=+-10Then simplify= x=8,-12

3) 9/x+2=x+2/4Cross multiply: 36=(x+2)2Square root: +-6=x+2Simplify: 6-2=4 and -6-2=-8x= 4, -8

How to check if proportion is equal

Just simply, cross multiply and see if the products are equal.

example1) 2:3=4:6In this proportion you first cross multiply: 12=12 So this is correct.

2) 10:5=3:4Do the same thing, cross multiply15=40, so therefore, this is incorrect.

3) 10:3=30:9So the answer is 90:90So therefore this is also correct.

Similar polygons

• Similar polygons are polygons that their corresponding angles are congruent, also there corresponding sides lengths are proportional.

Examples

These windows are similar polygons

These mountains are similar polygons

Examples

These mountains for ski, are similar polygons

Scale factors It tells you how much the picture has changed of size. It can get bigger or reduce of size.

Using similar triangles for indirect measurement

• When we measure huge things, that can’t be measured by rulers and any meter sticks, you can use sunlight to have indirect measurements. You step in the sun and see the shadow, measure the length of shadow from you and also measure the height of your height, then do the same thing with the object that you are trying to measure, place the height of that object as X and find the shadow from the tallest point of that object.

Examples

So the men wants to know the measurement of this tree. This man has height of 2m and the shadow of him was 1mThe shadow of the tree was 4m and the height of this tree is unknown so we place that as “X”. So we do the proportion to findThe height of this tree. 2/X=1/4, so when we cross multiply, it will be X= 8, so the height of this tree was 8meters.

So this men wants to know the height of this building, he knows that his height is 2meters and he realized that the shadow also measured 2meters, if the shadow of this building was 20meters what is the height of this building? So we use the proportion to solve this. 2/X=2/20, cross multiply which is 2X=40, so when we divide, the height of this building is 20meters

The men knows that his height is 5meters, and his shadow measured 3 meters this time, when we know that this dinosaur's shadow measured 10meters, what is the height of this dinosaur? 5/x=3/10, so when we cross multiply, 3x=50, when we divide it will be x=50/3 meters.

Trigonometric Ratios

• Ratio 1: Sine(sin): opposite side/hypotenuse• The sine of any angles in the triangle can’t be

more than 1.

a

A10

15

A

5 7A

9

13

3/2=1.5

7/5= 1.4

9/13=0.69

cosine

• Ratio 2: Cosine (Cos): adjacent side/hypotenuse

• It can’t have more than 1

A11 15

A5

6

A5 1611/15=0.73

5/6=0.83

5/16=0.3125

Using scale factor to find perimeter and area

• Since you are given the lengths of the triangle and a fraction that tells you how much it is enlarged or reduced, it can get bigger or smaller, you multiply the lengths times that fraction to get the new sides. You then add all the sides to get the perimeter.

• In the case of area, as the new figure is formed with given lengths, you just have to use area formulas for each corresponding figures.

1110

5

Let’s say that we are given these lengths and the fraction is 1/2, so we multiply the length with ½, which is 5.5, 5 , 2.5. to find perimeter of this new triangle, you just have to add them up which is 13.

12

6

In the rectangle, we are given with this lengths, and the fraction that we are going to multiply is 1/3, so we multiply 1/3 with (12 x 2) and 1/3 with (6 x 2) which is 8 + 4= 12

The area for this would be, 10/2 times 5/2 times ½ which is 6.25

The area for this new rectangle will be 12/3 times 6/3 which is 8

10

136

The triangle is given with these lengths, when we have new fraction of 1/4 we have to multiply each side with 1/4, so 13/4 + 10/4 + 6/4 = 1.5

In the case of Area, you multiply 6/4 with 10/4 and multiply that # with 1/2 so the answer is 0.5.

Tangent

• Ratio 3: Tangent (Tan): opposite side/adjacent side

• This is different from Sin and Cos, Tan can have a number that is more than 1.

A A

A

11

12

13

17

5

3

12/11=1.09

5/3=1.6

13/17=0.76

Solving a right triangle

• Solving a right triangle means to find all three lengths of the sides and all of three angles.

How to solve a right triangle using trigonometric ratios

• To find the length of a side:• Write a ratio that can be written with the info

you have, you may use Sin, Cos, or Tin to find the missing lengths of sides or angles, so you have to choose which ratio you will use.

• Leave the side you want to find alone• Solve

Sin42 = x/1212(sin42) = x

X = 8.02

x

42

12

Tan56=X/2626Tan56=X38.55=XX

5626

X

1416

xSin(14/16)xSin-1(14/16)x= 61

Angle of Elevation

• Angle of elevation: A angle formed by a horizontal line and a line of sight to a point above the line. (Watch from down to up.)

• Angle of depression: A angle formed by a horizontal line and a line of sight to a point below the line. (Watch from up to down.)

Angle of depression

Angle of elevation

Angle of elevation

Angle of depression

Angle of depression

Angle of elevation