A ratio is a quotient of two numbers. The ratio of two
numbers, a & b can be written as a to b, a:b, or a/b, (where
b = 0).
Examples:
7
1 to 2
1:2
1/2
3 to 4
3:4
3/4
7 to 8
7:8
7/8
3
Ratioof widht and7length is =
3:7:3:7
9
3
4
7
Ratio= 4:7:3:9
3
The ratios of a quadrilateral are 2:3:5:7. its
perimeter is 85 ft. What is the longest side?
-- 2x+3x+5x+7x =85.
-- 17x= 85
--x= 5
So the length of the longest side is 7(5) = 35 ft.
A proportion is an equation stating that two ratios are
equal .
Examples: 1/2=5/10
8:10 = 4:5
3/4 = 9/12
In the proportion a/b= c/d, the values a and d are the
extremes. The values b and c are the means. When the
proportion a:b=c:d—extremes are in the first and last
positions. The means are in the two middle positions.
A proportion and ratios are related because a proprotion
is the equality of tow ratios.
25/100= 1/4
1/50= 2/100
1/5= 3/15
In the proportion a/b= c/d, the product of extremes ad and
the product of the means bc are called cross products.
Property.
In the proportion a/b= c/d and b and d = 0, then ad = bc.
Examples:
5/y=45/63
5(63)= y(45)
315=45y
Y=7
x+2/6=24/x+2
(x+2)²= 144 (square root both sides)
x+2= ±12
x=10 or x = -14
2/9= 8/y
2(y)=8(9)
2y=72
y=36
x²/18=x/6
6x²=18x
x
6x=18
x=3
x/3= 4/6
X(6)=12
X= 2
This is how to solve a proportions by cross multiplication . To check if
you are correct after solving it, you plug in the number and solve again.
x/10= 18/20
X(20)=180
X= 9
**check
9/10=18/20
9*20=10*18
180=180
Algebra
x/6= 10/12
X(12)=60
X= 5
**check
5/6=10/12
5*12=6*10
60=60
x/3= 8/24
X(24)=24
X= 1
**check
1/3=8/24
1*24=3*8
24=24
Properties of Proportions.
The proportion a/b=c/d is
quivalent to the following:
ad=bc
b/a=d/c
a/c=b/d
Numbers
The proportion 1/3=2/6 is
equivalent to the following.
1(6)=3(2)
3/1=6/2
1/2=3/6
615.3 is the new height
4x=10y
x/y=10/4
x/y=5/2
......
16s=20t
s/t=16/20
s/t= 4/5
5x=15y
x/y=15/5
x/y=3/1
First of all, figures that are similar(~) have the same shape but not
necessarily the same size.
1 is similar to 2( 1~ 2)
1 is not similar to 2( 1~
2)
Similar Polygons
--Two polygons are similar if and only if their corresponding angles are
congruent and their corresponding side lengths are proportional.
Extra:
Similarity ratio: is the ratio of the lengths of the corresponding D
sides of two similar polygons.
A
6
6
The similarity ratio of ABC to DEF is 3/6, or ½. 3 3
The similarity ratio of DEF to ABC is 6/3, or 2. B
C E 6 F
3
Examples of similar polygons:
Describes how much the figure is enlarged or reduced.
First of all indirect measuring allows you to use properties of
similar polygons to find distances that are difficult to measure
directly.
The first step is to choose a tall object, like a tree or something
similar.
After choosing the object, grab a mirror and set it on the ground
facing the tall object and get away until you see the highest point
of the object.
Then, you measure the distance from where your standing at to
the mirror. Then your measure from the mirror to the tree. Finally
you measure your height and using all the measurements you set
a proportion and find the height of a tall object.
There is another type to measure a tall object, first you find the
tall object and measure the distance from the object to the
shadow. Then using your height and measuring your shadow you
form a proportion and find the height of the tall object.
This is an important subject to learn because you can use it in real
life. For example in architecture, when designing a house or any
work. Also another example when you are planning to cut a tree
down. If you cant measure directly you could use indirect
measuring to see how tall it is. After knowing its height you could
see if it hits your house or something around it.
The scale factor is helpful to find the perimeter and area of two
similar figures.
Perimeter:
If you want to find the perimeter of one figure and know the scale
factor between them, you first need to find the perimeter of each
triangle. Then you create a fraction with each perimeter(smaller
figure to bigger figure).Finally you simplify the fraction.
Area:
First need to find the area of each shape. You make a fraction using
the areas (small figure to big figure). Finally you simplify it and then
you square it.
10
Perimeter of figure 1: 40
Perimeter of figure 2: 20
So it is 20/40=1/2
5
10
1
10
5
2
5
10
1
12
12
4
9
8
10
6
Area of figure 1: 100
Area of figure 2: 25
So it is 25/100= (1/4)²
Perimeter of Figure 1: 30
Perimeter of Figure 2: 10
So it is 30/10=3/1
2 4
3
Area of figure 1: 27
Area of figure 2: 3
So it is 27/3=(9/1)²
2
6
5
5
4
3
Perimeter of figure 1: 24
Perimeter of figure 2: 12
So it is 24/12= 2/1
Area of figure 1: 24
Area of fiugre 2: 6
So it is 24/6=1/4=(1/2)²
First of all a trigonometric ratio is a ratio of two sides of a right
triangle.
The three basic trigonometric ratios are: Sin A = opp/ hyp,
cos A = adj / hyp and tan A = opp/adj.
These three ratios can be used any type of right triangle based
on what is given. Suppose you have an angle and know the
hypotenuse of it and they are asking to find the opposite side
to the given angle, than you have to use the Sin because it
relates opposite side and the hypotenuse.
Whenever they ask you to solve a triangle you have to find all
the measures of the triangle. That means two angles (ne is the
right) two legs and the hypotenuse.
The angle of elevation is when the measurement is
taking from the floor looking up to an specific point.
While the angle of depression is when the observer is
at a certain height and his looking down to an specific
point.
These two types of angles are used to solve problems
relating right triangles and need to find an unknown
measurement. In this type of problems you have to
know either angle and one leg or the hypotenuse to
find the other measurements of the right triangle
formed.
Angle of Depression
Angle of Elevation
Since horizontal lines are parallel,<1 = <2 by the
Alternate Interior Angles Theorem. Therefore
the < of elevation from one point is congruent
to the angle of depression from the other point.
_____(0-10 pts) Describe a ratio. Describe a proportion. How are they related? Describe how
to solve a proportion. Describe how to check if a proportion is equal. Give 3 examples of
each.
_____(0-10 pts) Describe what it means for two polygons to be similar. What is a scale
factor? Give at least 3 examples of each.
_____(0-10 pts) Describe how to use similar triangles to perform an indirect measurement.
Why is this an important skill? Give at least 3 examples.
_____(0-10 pts) Describe how to use the scale factor to find the perimeter and area of a new
similar figure. Give 3 examples of each, 3 for perimeter, 3 for area.
_____(0-10 pts.) Describe the three trigonometric ratios. Explain how they can be used to
solve a right triangle. What does it mean to solve a triangle? Give at least 3 examples of
each.
_____(0-10 pts.) Compare an angle of elevation with an angle of depression. How are each
used? Give at least 3 examples of each.