Download Chapter 7 Review

Right Triangles
Test Review
11-12-13
Pythagorean Theorem, Special
angles, and Trig Triangles
Pythagorean Theorem
Terminology
• We call the two sides that touch the right
angle the LEGS. They are the a and b.
• The side opposite the right angle is
the hypotenuse. It is the c.
• Label each part.
Terminology
• The hypotenuse is always the longest side
but shorter than the two legs put together.
• Here is the 60 Second Review
Steps to Solve
• If you know the 2
legs – square the
numbers – addtake square root.
Steps to Solve
• If you know the
hypotenuse and
one leg – square
the numbers subtract – take the
square root.
Special Right Angles
45 – 45- 90
Special Right Angles
If we represent the legs of an
isosceles right triangle by 1, we can
use the Pythagorean Theorem to
establish pattern relationships
between the lengths of the legs and
the hypotenuse. These relationships
will be stated as "short cut formulas"
that will allow us to quickly arrive at
answers regarding side lengths
without applying trigonometric
functions, or other means.
There are two pattern formulas that
apply ONLYto the 45º-45º-90º
triangle.
Special Right Angles
Special Right Angles
What if I forget the pattern formulas?
What should I do?
Let's look at 3 solutions to this problem where
you are asked to find x:
Pattern Formula solution
We are looking for the
hypotenuse so we will
use the pattern formula
that will give the answer
for the hypotenuse:
Substituting the leg = 7,
we arrive at the answer:
Pythagorean Theorem
Trigonometric solution
solution
Since a 45º-45º-90º, also Use either 45º angle as
called an isosceles right the reference angle. One
triangle, has two legs
possible solution is
equal, we know that the
shown below:
other leg also has a
rounded
length of 7 units.
rounded
c2 = a2 + b2
x2 = 72 +72
x2 = 49 + 49
x2 = 98
A nice feature of the
pattern formulas is that
the answer is alreadyPythagorean
in
Theorem, Special
reduced form.
angles, and Trig Triangles
Special Right Angles
Special Right Angles
If you draw an altitude in an equilateral triangle, you will form
two congruent 30º- 60º- 90º triangles. Starting with the sides
of the equilateral triangle to be 2, the Pythagorean Theorem
will allow us to establish pattern relationships between the
sides of a 30º- 60º- 90º triangle. These relationships will be
stated here as "short cut formulas" that will allow us to quickly
arrive at answers regarding side lengths without applying
trigonometric functions, or other means.
There are three pattern relationships that we can establish
thatapply ONLY to a 30º-60º-90º triangle.
Special Right Angles
Labeling:
H = hypotenuse
LL = long leg (across from 60º)
SL = short leg (across from 30º)
Again 1st step is to LABEL what you have!!
Short Cut Pattern Formulas
short leg:
hypotenuse:
H = 2SL
Long leg:
You must remember that
these formula patterns
can be used ONLY in a
30º-60º-90º triangle.
Easy Practice
x is the short leg
y is the long leg
LL = SL
Answer
Answer
Harder Practice
6 is the short leg and
x is the hypotenuse
(start with what you have
given)
H = 2 (SL)
y is the long leg
LL = SL
x = 2 (6)
Answer
Answer
y=6
Hardest Practice ( bonus?)
8 is the long leg and
x is the hypotenuse
(start with what you
have given)
y is the short leg
Answer
Answer
x =16/3
Right Δ Trigonometry
Right Δ Trigonometry
MUST LABEL SIDES FROM
ANGLE POINT OF VIEW
Label Hypotenuse 1st. Then label side opposite of angle. The side left is your adjacent side.
Right Δ Trigonometry
Right Δ Trigonometry
With both the TI 83 and 84 you need to make sure
you are in degrees.
With yellow TI 84, you need a ^^ after the angles to
tell the calculator you are in degrees. But you do
not need anything to solve for the inverse
operations. Zoom equation takes care of it.
Right Δ Trigonometry
There are actually 6 functions with trignonmetry.
However, the cosecant, secant, and cotangent are
just the reciprocal of the sine, cosine, and tangent.
When we do not know the degree of the angle, we
can calculate the fraction ratio, then find the
inverse of the ratio with the 2nd button on the
calculator. This isn’t needed in zoom 400 and
above. But in 300, divide the fraction, push 2nd then
your 9 sine, cosine, or tangent button, then 2nd and
(-) button for the answer of the fraction above.
Angle of Elevation
The angle of elevation is always
measured from the ground up. Think of it
like an elevator that only goes up. It is
always INSIDE the triangle.
In the diagram at the left, x marks the
angle of elevation of the top of the tree as
seen from a point on the ground.
You can think of the angle of elevation in
relation to the movement of your
eyes. You are looking straight ahead and
you must raise (elevate) your eyes to see
the top of the tree.
Angle of Depression
The angle of depression is always OUTSIDE the triangle. It is never inside the triangle.
In the diagram at the left, x marks the angle of depression of a boat at sea from the top of a
lighthouse.
You can think of the angle of depression in relation to the movement of your eyes. You are
standing at the top of the lighthouse and you are looking straight ahead. You must lower
(depress) your eyes to see the boat in the water.
As seen in the diagram above of angle of
depression, the dark black horizontal line
is parallel to side CA of triangle
ABC. This forms what are called
alternate interior angles which are equal
in measure (so, x also equals the
measure of <BAC).
Simply stated, this means that:
the angle of elevation = the angle of depression.
Let’s Practice
Let’s Practice
Let’s Practice
Let’s Practice
Things to Remember