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G E O M E T R Y
CHAPTER 9
RIGHT TRIANGLES AND
TRIGONOMETRY
Notes & Study Guide
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TABLE OF CONTENTS
SIMILAR RIGHT TRIANGLES ............................................................................. 3
THE PYTHAGOREAN THEOREM ...................................................................... 4
SPECIAL RIGHT TRIANGLES ............................................................................ 5
TRIGONOMETRIC RATIOS ................................................................................ 7
SOLVING RIGHT TRIANGLES ......................................................................... 11
HOMEWORK EXAMPLES ................................................................................ 13
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SIMILAR RIGHT TRIANGLES
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One way that two triangles were similar was by the AA Postulate (2 angles from
one congruent to 2 angles from another). From Chapter 8
For right triangles, it is possible to create three similar triangles (all proven by
AA) by just adding one altitude. This is Theorem 9.1
Since the triangles are all similar, all angles are equal and all sides are
proportional. To find any missing info, work the same way as Chapter 8.
RECOMMENDED TECHNIQUE
Separate the three triangles (makes identification easier)
Create your proportion (be consistent with pieces you use)
Use cross products to solve as usual.
EXAMPLE
Set up your proportion
and do cross products
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Separate the triangles for easy identification
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THE PYTHAGOREAN THEOREM
In all right triangles, there is a pattern involving the lengths of the sides. The
pattern is represented by one of the most recognized formulas in mathematics.
PYTHAGOREAN THEOREM – In a right triangle,
the square of the hypotenuse is equal to the sum
of the squares of the other two sides.
a2 + b2 = c2 (c is the hypotenuse)
EXAMPLES
PYTHAGOREAN TRIPLES
Any three integers (no decimals) that make Pythagorean’s Theorem true is called
a Pythagorean Triple.
Triples are shortcuts...if you recognize that one is there, you don’t have to use
the formula.
MOST COMMON TRIPLES: 3, 4 and 5 5, 12 and 13 7, 24 and 25
CLASSIFYING TRIANGLES USING THE THEOREM
Pythagoreans Theorem can be used to determine what type (acute/right/obtuse)
of triangle that you have. You do it by comparing the c2 part to the a2 + b2 part.
c 2 < a 2 + b2
c 2 = a 2 + b2
c 2 > a 2 + b2
acute
right
obtuse
Substitute the value of the sides into the formula and see which one occurs.
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SPECIAL RIGHT TRIANGLES
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All right triangles fit the Pythagorean Theorem pattern (a2 + b2 = c2). But…
…there are two specific right triangles, called SPECIAL RIGHT TRIANGLES,
that have another pattern in themselves. They are identified by their angle sizes.
30–60–90
45–45–90
Pythagorean’s Theorem requires you know 2 sides to use it. Special Right
Triangles only require one…that’s the shortcut.
30–60–90 TRIANGLE
Short leg – the side opposite the 30° angle
Long leg – the side opposite the 60° angle
Hypotenuse – the side opposite the 90° angle
Hypotenuse = 2 • short leg Long leg = √3 • short leg
The 30-60-90 pattern is based on the short leg!
45–45–90 TRIANGLE
Legs – the sides opposite the 45° angles
Hypotenuse – the side opposite the 90° angle
Hypotenuse = √2 • Leg
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Legs are equal
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RIGHT TRIANGLE EXAMPLES
Find the values of the missing sides of each right triangle.
Use Pythagorean’s Theorem to classify each triangle.
Find the unknown side length in each right triangle. Round off any answers to 2
decimal places.
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TRIGONOMETRIC RATIOS
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– All right triangles work with Pythagorean’s Theorem, but you need 2 sides to use it.
– If you don’t have 2 sides, special right triangles are used, but they are very specific.
– If you still don’t have 2 sides and no special right triangle, that’s where trig comes in.
A trigonometric (trig) ratio is a ratio (fraction) that compares any two sides from
a right triangle. For every different pair of sides used, you get a different trig ratio.
The sides are named by their position to the specific angle being referenced.
EXAMPLE: Angle A is the reference angle.
Opposite  directly across from the angle
Adjacent  next to the angle
Hypotenuse  across from right angle
DEFINING THE TRIG RATIOS
There are six different ways to create a ratio using any two sides from the right
triangle. Each way is a trig ratio. The three primary ratios we will use are…
EXAMPLE
sin A =
cos A =
tan A =
sin B =
cos B =
tan B =
Trig ratios are usually written as decimals to four decimal places!
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TRIGONOMETRIC RATIOS
HOW TRIG RATIOS RELATE TO ANGLES
It is important to know that all trig ratios are unique. This means that every
individual trig ratio corresponds to just one angle size.
So when you find a trig ratio, that value can only exist for one specific size of
angle. If the ratio changes, then the angle size does too.
EXAMPLE:
Say we find that sin A = 0.6429, that ratio means A = 40°
Say we find that cos B = 0.4500, that ratio means B = 63.3°
Say we find that tan C = 0.5317, that ratio means C = 28°
How do we know? Two ways…
(1) we use a scientific calculator
(2) we use a TABLE OF TRIG VALUES that has all the ratios matched up
to their angle sizes. It’s on page 845 of your textbook.
EXAMPLE 1
What is the sin A? _______
According to the table that ratio means A = ______
What is the size of angle B? _______
(use any trig ratio you choose)
EXAMPLE 2
Find the sizes of angles R and S.
R = ______
S = ______
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SOLVING RIGHT TRIANGLES
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Every right triangle has the follow pieces…
3 angles (1 right, 2 acute)
3 sides (1 hypotenuse and 2 legs)
Finding out the measure of all 3 angles and the lengths of all 3 sides is what is
referred to as SOLVING A RIGHT TRIANGLE.
HOW TO SOLVE A RIGHT TRIANGLE
To solve a right triangle, you will make use of two primary techniques…
(1) Pythagorean’s Theorem (used to find sides)
(2) Trig Ratios (used to find angles)
In most cases, you will need to use both methods (one time each) to
completely solve a right triangle.
EXAMPLE 1
Solve the right triangle shown.
g = ______ h = _______ angle G = _______
EXAMPLE 2
Solve the right triangle shown.
c = ______ angle A = _______ angle B = _______
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HOMEWORK EXAMPLES
HOMEWORK EXAMPLES
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HOMEWORK EXAMPLES