Right Triangles and Trigonometry

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Right Triangles and Trigonometry

• Chapter 8

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8.1 Geometric Mean

• Geometric mean
• Ex Find the geometric mean between 5 and 45

• Ex Find the geometric mean between 8 and 10

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• 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
C
D
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• The altitude from a right angle of a right
  triangle is the geometric mean of the two
  hypotenuse segments

B
Ex
A
C
D
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• The leg of the triangle is the geometric mean of
  the hypotenuse and the segment of the hypotenuse
  adjacent

B
Ex
A
C
D
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Find c, d, and e.
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Find e and f . (round to the nearest tenth if
necessary)
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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

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Pythagorean Theorem
c
a2 b2 lt c2 obtuse a2 b2 gt c2 acute
a
b

• When c is unknown

• When a or b is unknown

x
5
14
7
3
x
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• 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
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A. Find x.
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B. Find x.
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• 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.

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• 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.

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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
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• 45-45-90
• The legs are congruent

Ex
Ex
6
x
x
x
8
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A.
B.
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Find x and y.
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Find x and y.
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Find x and y.
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8.4 Trigonometry In Right triangles
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• A. Express sin L, cos L, and tan L as a fraction
  and as a decimal to the nearest ten thousandth.

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Find the value to the ten thousandth.

• Sin 15
• Tan 67
• Cos 89.6

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Find the measure of each angle to the nearest
tenth of a degree

• Cos T .3482
• Tan R .5555
• Sin P .6103

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Find y.
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Find the height of the triangle.
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• 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.
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Find angle D.
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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?

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• 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?

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• 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?

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• 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?

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• 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?