Transcript JOURNAL 7,8

JOURNAL 7,8
• A ratio is when you express two
numbers that are compared by division.
• It can be shown in different ways:
A:B
A to B
A/B
Examples:
The ratio of studentes per teacher is 4:1, 4
to 1, 4/1.
The ratio of tennis studemts per
court is 6:1, 6 to 1, 6/1.
The ratio of the peeople in red per
people in white is 53 to 4, 53:4, 53/4
• A proportion is an equation that states
that two ratios are the same, equal.
• How to solve a proportion:
1. First you have to cross multiply the
numbers and the variable. After you are
done multiplying, you have to leave
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.
Example 1:
x/5 = 4/10
10x=20
/10 /10
x=2
Examples 2:
6/x = 3/8
3x=48
/3 /3
x=16
Examples 3:
x/12 = 6/7
7x=72
/7 /7
10.3/12=6/7
x=10.3
2/5=2/5
6/16=4/10
• A ratio and a proportion are related
because a proportion is an equation
where 2 ratios are equal. Also because
they depend on each other, if you want
to make a proportion you need to have
ratios.
* If you want to make sure your
proportion is equal 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
• When two polygons are similar, it means
that their corresponding angles are
congruent and their corresponding side
lengths are proportional.
Examples:
10
10
5
8
5
2
8
2
1
8
X has to be 4!
x
X has to equal 20!
x
X has to equal 16!
4
16
16
x
4
5
• A scale factor describes how much a
figure can be reduced or enlarged. A
ratio of two corresponding lengths in
d
two figures.
a
2
a
2
6 f
6
d
4
4
g
2
20
20
e
b
c
2
6
b
The ratio of ab to df is 6:2
10
c
The ratio of bc to ef is 10:2, 5:1
f
d
a
9
18
b
18
14
e
c
The ratio of ac to df is 18:9, 2:1
9
7
f
• When using similar triangles to perform
indirect measures you have to use
proportions to find your answer. Use
the actual height and the measure of
shadows.
• It is an important skill because with it,
in real life you could find many
measurments of things like builidings,
houses and it helps in different jobs.
The boy wants to see how high his
house is because he wants to see if he could
climb it with his friends some day. The shadow
of his house is 5. His hight is 4.5 meters and
his shadow is 2.
x
4.3
2
5
x
4.3
20
2
5/x = 4.3/2
4.3x=10
/4.3 /4.3
X=2.3
The same boy is very curious and wants
to know the hight of the statue of liberty
becuase he didnt find that sort of
information in his textbook. He knew
that that the shadow of the statue was
20 and his hight was 4.3 and his shadow
was 2.
20/x = 4.3/2
4.3x=40
X=9.3
33/x = 4.3/2
4.3x=66
X=15.3
x
4.3
33
The same boy went to paris and wanted to know
the height of the eifel tower since his parents
didnt tell him. He heard in class that its shadow
was 33. he knew that his height was 4.3 and his
shadow was 2.
2
How to use the scale factor to find the
perimeter and area.
• PERIMETER: To find the perimeter using scale
factor, you first need to find the perimeter of
each triangle. Then you create a fraction with
each perimeter, but remember, it is the smaller
shape over the bigger shape. Then you simplify
the fraction. The ratio of the perimeter is the
same as the ratio of their sides.
• AREA: To find the area using scale factor, you
first need to find the area of each shape. You
make a fraction out of those areas (small shape
over the bigger shape). After you have made
the fraction, you simplify it all the way and
then you square it.
10
4
Perimeter: 12/24 = 1/2
5
Area: 12*2/48*2 = 1*2/4*2
8
3
4
6
6
4
4
Perimeter: 16/24 = 2/3
Area: 16*2/ 36*2 = 4*2/ 9*2
6
4
6
6
6
3
3
6
Perimeter: 8/17
Area: 2.5*2/13.75*2
2
5
The three trigonometric ratios
•
•
•
•
Sin=Opposite/Hypotenuse
Cos=Adjacent side/Hypotenuse
Tan=Opposite side/Adjacent
To solve a triangle is to find all the
sides and angles of a triangle.
8
Sin: 7/9
Cos: 5/9
Tan: 7/5
Sin: 8/12
Cos: 6/12
Tan: 8/6
12
9
7
6
5
Sin: 5/15
Cos: 13/15
Tan: 5/13
15
5
13
How are they used to solve for
right triangles?
• To find the angles:
1. you need to know what type of ratio it is.
2.Then you plug in the the inverse of the ratio into
your calculator (Ex. Sin-1(12/13) = 66.92)
• To find the sides:
1.To find the side, you need to find the type of ratio.
2.Then, you make a proportions with the missing value
and the other side.
3.You plug in the side measure and then the ratio with
the angle measure between those sides. (Ex. Tan40 =
X; 100(tan40) = 83.9
Sin42: x/12
12(sin42)=x
X= 8.02
12
x
42
Sin(12/13)
Sin-1(12/13)
x
12
X = 67.38
8
x
Sin42 = 8/x
8 = xSin42/sin42
42 X = 11.96
13
Angles of Elevation and
Depression
• Angle of Elevation: it is the 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: the angle formed
by a horizontal line and a line of sight
into a point below the line. Watch from
up to down.
Angle of elevation
Angle of depression