Download Journal 7

RATIO – a ratio compares two numbers by dividing. The
ratio of two numbers can be written in various ways such as
a to b, a:b, or a/b, where b doesn’t equal 0. For example
the ratios of 3 to 4 can be represented as 3:4 or ¾
PROPORTION – a proportion is an equation stating that
two ratios are equal. In this proportion a/b = c/d, a and d
are the extremes when b and c are the means. When a
proportion is written as a:b = c:d, the extremes are in the
first and last positions, which means that the means are in
the two middle positions.
Proportions and ratios are related since proportions use ratios so inevitably, they both
compare two number by division and can be used to create detailed miniature models.
To solve a proportion, you need to use
the Cross Products Property (in a
proportion, if a/b = c/d and b and d
don’t = 0, then ad = bc) so according
to the Properties of Proportions, the
proportion a/b = c/d is equivalent to
the following:
•ad = bc
•b/a =d/c
•a/c = b/d
EX1.
3:4 = 12:x
3x = 48
x=16
EX2.
5/17 = x/6
30 = 17x
x = 1.76
EX3.
297 to 2 = x to 1
297 = 2x
x = 148.5
Two polygons are similar polygons iff their
corresponding angles are congruent and their
corresponding side lengths are proportional. For
them to be similar can also mean that they have
the same shape but not necessarily the same size.
- a scale factor is for describing how much a figure is
enlarged or deduced. For dilation, transformation that changes size of a figure but
not shape, with scale factor k, you can find the image of a point by multiplying each
coordinate by k: (a,b)  (ka, kb)
4
90
90
2
90
EX1
6
90
15
18
EX2
2
3
EX1
90
90
15
90
90
15
7.5
3
9
EX2
7.5
EX3
25
1
45
9
EX3
75
To find indirect measures* with similar triangles, you have to follow
some steps:
1. Convert the measurements to a single unit (if needed)
2. Find the similar triangles
3. Find the measurement you need... (use cross products property)
This skill is important because if you were cutting down a tree
near your house and you want to calculate if it is safe to do so,
you may use this skill to find out the height of the tree.
*Indirect measures are any method that uses formulas, similar figures, or proportions to measure an object.
EXAMPLE 1
¾ = 20/h
3h = 80
h = 26.667
4ft
3ft
20ft
h
EXAMPLE 2
4/5.3 = 29/h
4h = 153.7
h = 38.425
5.3ft
4ft
29ft
h
EXAMPLE 3
h/6 = 40/136
136h = 240
h = 1.765
6ft
h
40ft
136ft
10+5+15/30+20+10
30/60
=1/2
2(4+3)/2(9+12)
7/21
=1/3
4
30
3
9
15
20
12
10
EX2
Circumference 75
10
EX3
5
EX3
Circumference 37
75/37
Already in simplest form…
B
c
A
a
C
b
1. SINE – the sine of an angle is the ratio of the length of the leg opposite the angle to
the length of the hypotenuse.
• sinA = opposite leg/hypotenuse = a/c
• sinB = opposite leg/hypotenuse = b/c
2. COSINE – the cosine of an angle is the ratio of the length of the leg adjacent to the
angle to the length of the hypotenuse.
• cosA = adjacent leg/hypotenuse = b/c
• cosB = adjacent leg/hypotenuse = a/c
3. TANGENT – the tangent of an angle is the ratio of the length of the leg opposite t he
angle to the length of the leg adjacent to the angle.
• tanA = opposite leg/adjacent leg = a/b
• tanB = opposite leg/adjacent leg = b/a
You can also reverse it into (sin-1, cos-1, or tan-1)
S.O.H.C.A.H.T.O.A.
SINE.OPPOSITE.HYPOTENUSE.COSINE.ADJACENT.HYPOTENUSE.TANGENT.OPPOSITE.ADJACENT
Since by the AA Similarity
Postulate, a right triangle with a
given acute angle is similar to
every other right triangle with the
same acute angle measure, and
since a trigonometric ratio is a
ratio of two sides of a right
triangle, this can help us solve
right triangles.
A
EX1.
SinB = 5/6
EX2.
CosB = 4/5
EX3.
TanB = 5/4
EX1.
Sin-1B = 5/6 = 56.443
EX2.
Tan-1B = 5/4 = 51.34
EX3.
Cos-1B = 36.87
6
5
B
C
4
ELEVATION – the angle of elevation is the angle formed by a
horizontal line and a line of sight to somewhere above the
line. This is important when you are in a watch tower
watching a plane descend in order to give correct directions.
DEPRESSION – the angle of depression is the angle formed
by a horizontal line and a line of sight to somewhere below
the line. This is important when you are a forest ranger and
you need to stand high in an observation tower in order to
see if a forest fire breaks out.
<of depression
<of elevation
<of depression
<of depression
<of elevation
<of elevation