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Vocabulary and Notes
Introduction to Chapter 8
Right Angle Triangles and
Trigonometry

Right angle triangles have two perpendicular legs that
create a right angle.

The other two angles can be of any measure.

However, we are most interested in right angle triangles
that have measures of either 30-60-90 or 45-45-90 for
their angles.
Legs of right triangles

Right angle triangles have a hypotenuse, which is
across from the right angle, and two other legs.

If the two legs are of different lengths they are referred
to as the “long leg” and “short leg” of the triangle.

Otherwise, when they are of equal lengths, they are
referred to as simply the “legs” of the triangle.
2
3
Hypotenuse comes
from the Greek words
for “under tension”,
meaning “stretched”.
Legs of right triangles

The ratios of the legs of the right triangles are
important.

The ratios define the relative lengths of each leg. That
is, the measure of each leg when compared to the other
legs.
2
3
Legs of right triangles

This means that if you know the measure of one leg
you can find the others by using these known ratios.

Because of this we often express the measure of the
legs in terms of “x”.
2𝑥
2𝑥
3𝑥
𝑥
𝑥
𝑥
The Pythagorean Theorem

The Pythagorean Theorem states that the square of
the hypotenuse is equal to the sum of the squares of
the two legs.

hypotenuse2 = (shorter leg)2 + (longer leg)2

hypotenuse2 = (leg)2 + (leg)2

Often shown as: a2 + b2 = c2 or x2 + y2 = z2
Working with radicals


Because the relationship between the legs and
hypotenuse involve terms that are squared we need to
take the square root to solve for unknowns.
The square root symbol, √ , is also known as the
radical.

We sometimes need to simplify the radical expression.
This means we reduce the term that involves the radical
to one that involves prime factors that cannot be
further reduced.

We do this by building factor trees, or a tree of the
factors that can be derived from the number.
Working with radicals

Examples!

16 The square root of sixteen can be simplified because it
has (prime) factors that can be expressed as perfect squares.
This is important as we can only remove pairs of numbers from
under the radical sign.

We can build a factor tree to find those factors

Each pair of numbers can be separated
out from the term under the radical
sign.

In the case of 16 we end up with two
pairs of “2”, so we can remove them as
“2” and “2” (each pair under the
radical is equal to a single number
outside the radical).

This leaves us with 2 x 2 or 4.
Working with radicals

Another example!

24 The square root of twenty-four can also be
simplified.

We again build a factor tree to find those factors

Again, each pair of numbers can be
separated out from the term under the
radical sign.

In the case of 24 we end up with one
pair of “2” and singles of “2” and “3”.

Since we can only remove pairs of
numbers from under the radical this
leaves us with 2 on the outside and 2 x 3
on the inside or 2 2𝑥3 which is 2 6.
Working with radicals

More examples!

20 How would we simplify this radical expression?

Step one: build a factor tree to find the prime factors

This time we have a pair of “2”s and a
single term of 5.

Since we can only remove pairs of
numbers from under the radical this
leaves us with 2 on the outside and 5 on
the inside or 2 5.

What if the number was 27? What would
the tree look like?
Your turn!

On a piece of paper simplify the following radicals.
Remember to build a factor tree for each one and
remove pairs of numbers from under the radical sign as
single numbers outside the radical sign.

18

32

25

50

28

40
Self Assessment: when you are done, write
down how you would rate yourself on working
with radicals from 1 to 4.
1 = I don’t understand; 2 = I kind of get it but still
need help; 3 = I can do this with just a little help;
4 = I have this down and can explain it to others.
Answers

18 simplifies to 2 x 3 x 3, so we can remove the pair
of “3”s, giving us 3 2 .

32 simplifies to 2 x 2 x 2 x 2 x 2, so we can remove
two pairs of “2”s, giving us 2 x 2 2 or 4 2 .

25 simplifies to 5 x 5 under the radical or just 5
outside the radical.

50 simplifies to 2 x 5 x 5, so we can remove the pair
of “5”s, leaving us with 5 2 .

28 simplifies to 2 x 2 x 7, so we can remove the pair
of “2”s, leaving us with 2 7 .

40 simplifies to 2 x 2 x 2 x 5. We can only remove the
one pair of “2”s, leaving us with 2 x 5 under the
radical. This yields 2 2𝑥5 or 2 10 .