Download chapter 7 and 8

Chapter
7 and 8
By: Ou Suk Kwon
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Comparing 2 numbers that are written:
A to B
A/B
A:B
The ratio of red
fruits to another
color fruits are 4:8
The ratio of dirty cars to clean cars is
1:2
The ratio of black guns to
white gun is 6:1
• Proportion is an equation that states that two
ratios are equal to each other. (compares ratio
stating that they are same.)
Proportion
9:2=18:4
Proportion
4:6=2:3
Proportion 1:2=3:6
how to solve proportions
If there are 2 variables
Cross multiply the fraction then,
square root both sides
Use + and – root to solve for x
The answer for X will be 2 numbers
If there is only one variable
Cross multiply the fraction and then
divide.
1) 16/2=X/4
First cross multiply so the answer will be 2x=64
Then divide which is X=32
2) 10/x+2=x+2/2
So here, cross multiply: (x+2)2=20
Then multiply which is x+2=+-10
Then simplify= x=8,-12
3) 9/x+2=x+2/4
Cross multiply: 36=(x+2)2
Square root: +-6=x+2
Simplify: 6-2=4 and -6-2=-8
x= 4, -8
Just simply, cross multiply and see if the
products are equal.
example
1) 2:3=4:6
In this
proportion you
first cross
multiply: 12=12
So this is correct.
2) 10:5=3:4
Do the same thing, cross multiply
15=40, so therefore, this is incorrect.
3) 10:3=30:9
So the answer is 90:90
So 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 1m
The 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 find
The 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
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
5/6=0.83
11/15=0.73
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.
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.
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
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 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.
12/11=1.09
13/17=0.76
5/3=1.6
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/12
12(sin42) = x
X = 8.02
Tan56=X/26
26Tan56=X
38.55=X
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 depression
Angle of elevation
Angle of
depression
Angle of
elevation