Transcript Right Triangles and Trigonometry

Right Triangles and
Trigonometry
Chapter 8
8.1 Geometric Mean
 Geometric mean:

Ex: Find the
geometric mean
between 5 and 45
# x

x #

Ex: Find the
geometric mean
between 8 and 10
 If an altitude is drawn from the right angle of a
right triangle. The two new triangles and the
original triangle are all similar.
B
A
D
C
ABC ~ ADB ~ BDC
 The altitude from a
right angle of a right
triangle is the
geometric mean of
the two hypotenuse
segments
Ex:
AD BD

BD DC
hyp.seg .1 altitude

altitude hyp.seg .2
B
A
D
C
 The leg of the
hypotenuse
leg
triangle is the

leg
adjacent.hyp.seg .
geometric mean of
the hypotenuse and
the segment of the
B
hypotenuse adjacent
Ex:
AC AB

AB AD
AC BC

BC DC
A
D
C
Find the geometric mean between
Find c, d, and e.
Find e and f .
(round to the nearest tenth if necessary)
8.2 Pythagorean Theorem and
its Converse
 When and why do you use the
Pythagorean Theorem?


When: given a right triangle and the length
of any two sides
Why: to find the length of one side of a
right triangle
 When do you use the Pythagorean
Theorem Converse?

When: you want to determine if a set of
sides will make a right triangle
Pythagorean Theorem:
c
a
a b  c
2
2
a2 + b2 < c2 obtuse
2
a2 + b2 > c2
acute
b
 When c is unknown:
 When a or b is unknown:
x
5
14
7
3
x
 Converse: the sum of
the squares of 2
sides of a triangle
equal the square of
the longest side

8, 15, 16
 Pythagorean Triple:

3 lengths with
measures that are all
whole numbers &
that always make a
right triangle
3, 4, 5
 5, 12, 13
 7, 24, 25
 9, 40, 41

Not =, so not a right triangle
A. Find x.
B. Find x.
A. Determine whether 9, 12, and 15 can
be the measures of the sides of a
triangle. If so, classify the triangle as
acute, right, or obtuse. Justify your
answer.
B. Determine whether 10, 11, and 13 can
be the measures of the sides of a
triangle. If so, classify the triangle as
acute, right, or obtuse. Justify your
answer.
8.3 Special Right Triangles
 30-60-90


Short leg is across from
the 30 degree angle
Long leg is across from
the 60 degree angle
Ex:
14
x
30
y
shortleg  3  longleg
shortleg  2  hypotenuse
 45-45-90

Ex:
x
The legs are
congruent
leg  2  hypotenuse
Ex:
6
x
x
8
A.
B.
Find x and y.
Find x and y.
Find x and y.
The length of the diagonal of a square is
centimeters. Find the perimeter of the
square.
8.4 Trigonometry In Right
triangles
A. Express sin L, cos L, and tan L as a
fraction and as a decimal to the nearest
ten thousandth.
Find the value to the ten
thousandth.
 Sin 15
 Tan 67
 Cos 89.6
Find the measure of each angle to
the nearest tenth of a degree
 Cos T = .3482
 Tan R = .5555
 Sin P = .6103
Find y.
Find the height of
the triangle.
 When you need to find the angle
measure- set up the problem like normal

Then hit the 2nd button next hit sin, cos or
tan (which ever you are using) then type in
the fraction as a division problem, hit =
Find angle P.
Find angle D.
8.5 Angles of Elevation and
Depression
 Draw a picture and solve using
trigonometry.

Mandy is at the top of the Mighty Screamer
roller coaster. Her friend Bryn is at the
bottom of the coaster waiting for the next
ride. If the angle of depression from Mandy
to Bryn is 26 degrees and The roller
coaster is 75 ft high, what is the distance
from Mandy to Bryn?
 Mitchell is at the top of the Bridger Peak
ski run. His brother Scott is looking up
from the ski lodge. If the angle of
elevation from Scott to Mitchell is 13
degrees and the ground distance from
Scott to Mitchell is 2000 ft, What is the
length of the ski run?
 An observer located 3 km from a rocket
launch site sees a rocket at an angle of
38 degrees. How high is the rocket at
that moment?
 A kite is flying at an angle of elevation of
40 degrees. All 50 m of string have been
let out. What is the height of the kite?
 Two buildings on opposite sides of the
street are 40 m apart. From the top of
the taller building, which is 185 m tall,
the angle of depression to the top of the
shorter building is 13 degrees. How high
is the shorter building?