Download Right Triangles

Unit 7
“Right Triangles”
YOU’RE RIGHT!
Academic Geometry
Fall 2013
Name_____________________________ Teacher__________________ Period______
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Unit 7 at a glance
“Right Triangles”
This unit focuses on the investigation of the relationships between and within right
triangles. Students will build on prior knowledge to solve for all missing measurements
within right triangles.
Essential Questions
 What strategies can I use to identify patterns and develop geometric relationships?
 How do concrete models help me understand mathematics?
 How does geometry explain or describe a structure within the real world? How can it
help me solve a problem?
In Unit 7, students will…
 Derive and justify the Pythagorean Theorem and apply the Pythagorean Theorem and
Pythagorean triples (including making connections to similar right triangles) to solve
problems in a variety of contexts;
 Extend the Pythagorean Theorem by applying the converse of the Pythagorean
Theorem to classify triangles using side lengths and solve problems in both real-world
and purely mathematical situations;
 Investigate using patterns, develop, apply and justify triangle similarity relationships
between the length of the attitude drawn to the hypotenuse of a right triangle and the
length of the hypotenuse;
 Apply and justify properties of altitudes of similar right triangles (including the
Geometric Mean (Altitude) Theorem and Geometric Mean (Leg) Theorem) to solve
problems in situations;
 Investigate using patterns, develop, apply and justify 45-45-90 and 30-60-90 as similar
triangles to solve problems in both real-world and purely mathematical situations;
 Develop, justify, and apply the tangent ratio to solve problems in both real-world and
purely mathematical situations;
 Develop, justify, and apply the sine and cosine ratios to solve problems in both realworld and purely mathematical situations including angle of elevation and angle of
depression.
 Apply trigonometric ratios to solve problems in both real-world and purely
mathematical situations including using inverse trigonometric functions when
appropriate.
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Vocabulary
acute triangle
geometric mean
altitude of a triangle
hypotenuse
angle of elevation
inverse
angle of depression
inverse cosine
converse
inverse sine
cosine
inverse tangent
denominator
irrational numbers
isosceles triangle
sine
leg of a right triangle
solving a right triangle
numerator
special right triangles
obtuse triangle
tangent
perfect square
trigonometric ratio
principal root
pythagorean theorem
radical
radicand
rationalizing
right triangle
similar polygons
simplest radical form
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Radicals (Day 1) NOTES
Simplifying Radicals
A radical is another term for “square root”. For example “radical 4” means the same as
“square root of 4”. So if you are going to simplify “radical 4”, then this is the same as
taking the “square root of 4”. If no symbol is written in front of the radical symbol, then a
positive answer is presumed. This is called the principal root. If you were solving a
quadratic equation in the form
then assume that both the positive and negative
solutions are desired. For example, the equation
has two possible solutions for
….they are 2 and -2. A calculator will only give you the positive answer. For these
problems you need to know that there is also a negative answer and include it as part of
your solution.
“radical 4” or
“square root of 4”
is 2
√
This equation has
two solutions, 10
and -10
When you are evaluating a radical that is a perfect square you get an integer, such as the
two examples above. Another example is √ , which is 9. When you evaluate a radical
that is not a perfect square, the result is a decimal. An example is √ . You get
6.32455532 (try it in your calculator). Even this is not an “exact” answer, since your
calculator only has the capacity to show you a certain amount of numbers. (This number
is called irrational.)
Sometimes we write the answers to radicals that are not perfect squares with more
radicals. This is called “simplifying a radical”. The object is to get a number under the
radical sign that does not have a perfect square factor.
Simplify 40 ….an easy way to do this is to factor it into all its prime factors (make a factor
tree).
6
Another example………Simplify √
.
Whenever you get factors that are “doubles”, all the doubles will multiply to a
perfect square which has an integer square root. When there are no more
“doubles”, you’re done and whatever is leftover stays under the radical. As a
shortcut, you could also factor out the largest prime number factor, if you
immediately know it.
EXAMPLE PROBLEMS
Simplify the radicals.
1. √
2. √
3. √
5. √
6. √
TRY IT WITH VARIABLES!
4. √
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Do you see a SHORTCUT to simplifying radicals with variables?
One application of simplifying radicals is in solving quadratic equations of the type
with x2 = c. (You will see this again when solving Pythagorean Theorem problems.)
Write each solution as a simplified radical. Remember to include all possible solutions.
7.
8.
9.
Multiplying Radicals
When multiplying radicals, simply multiply the radicands (the terms under the radical
signs) together to get your new result under a new radical sign. Then simplify as you have
previously learned.
Examples:
1.
2 2 
2 2 
2.
5  40 
5  40 

4.  3 5 
3.

8x 
12 x
2
4  _____
____  __________
_____ _____ 
_______  __________
 ________________________________________________
***Helpful hint: When multiplying terms with variables, you add the exponents of the like variables that
are being multiplied together. x = x1……..so x ∙ x = x1 ∙ x1 = x 1+1 = x2.
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Simplifying Radicals with Fractions (Part 1…perfect square in denominator)
General Rule: √
√
√
Examples:
1.
4

81
4

81
2.
6

6
49m
3.
24
 __________________________
200
___________
 ____________
Careful! Reduce
first!
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Radicals (Day 1) HOMEWORK
Simplify the Expression.
1) √
2) √
3)
4) √
5) √
6) √
7) √
√
8) √
√
√
9) √
10
10) √
11) √
12)
13) √
14)
15) √
16) √
√
17)
√
√
√
√
18) ( √ ) √
Solve each equation. Remember to include all possible solutions.
19)
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20)
21)
Radicals (Day 2) NOTES
Simplifying Radicals
Simplifying Radicals with Fractions (Part 2…Rationalizing the Denominator)
Division of radicals:
The square root of the quotient is equal to the quotient of the square roots.
√
√
√
Having a radical in the denominator is a big
“no-no”! You will have to rationalize!
Rationalizing the denominator means to remove all radicals in the denominator of a
fraction.
 Simplify, if possible.
 Multiply the fraction by 1, but in the form of the fraction using the radical in the
denominator as both the numerator and denominator.
 Simplify, if possible.
This is equivalent to multiplying by 1, since
√
√
.
1.
√
√
√
√
√
√
√
√
√
√
It is okay to have a radical in the numerator.
This radical has been simplified.
EXAMPLE PROBLEMS
1)
3
5
2)
18
2
3)
8
3
12
4)
5
15
5)
20 2
15
6)
4
24
Solve for x. Simplify.
Leave your answer in exact form (as a radical, not a rounded decimal).
***SPECIAL NOTE: It is customary to write the variable in front of the radical.
For example, write “ √ ” instead of “√ ” to avoid confusion.
x
4
=
7)
2
3
10)
13
2
3
=
2
x
8)
5 3

2
x
9)
11)
6
3

2
x
12)
4
x
=
x
10
1
2
=
x
5 2
Radicals (Day 2) HOMEWORK
Simplify each expression.
1)
4)
2)
√
√
5)
√
√
√
3)
6)
√
√
Solve for x. Simplify.
7)
√
8)
√
9)
√
√
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Mixed Problems. Simplify.
10)
13)
15
√
√
11)
√
12) √
√
14) √
√
15)
√
7.1/7.2 NOTES
Pythagorean Theorem & Converse
The Pythagorean Theorem can be used to find the lengths of the sides of a right triangle.
Pythagorean Theorem: In a right triangle, the square of the lengths of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.
c2  a 2  b2
 hyp 
2
 leg    leg 
2
2
EX: Find the length of the unknown side in each right triangle.
Write the answer as a radical in simplified form.
(1)
(2)
EX: Find the lengths of the sides of a square that has a diagonal length of 42 mm.
Write as both a radical and decimal answer.
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Pythagorean Theorem:
IF it’s a right triangle, THEN
c2  a2  b2
Converse of the Pythagorean Theorem:
In any triangle, if c is the largest side and a and b are the other two sides,
2
IF c
a2  b2
, THEN it’s a right triangle.
2
IF c
a2  b2
, THEN it’s an acute triangle.
2
IF c
a2  b2
, THEN it’s an obtuse triangle.
EX: Determine if the side lengths represent those of an acute, right, or obtuse triangle.
(3) √
, 24, 30
(4) 14, 7, √
(5) 8, 10, 12
A Pythagorean triple is a set of three positive integers a, b, and c, that satisfy the equation
c2  a2  b2 . Other triples are found by multiplying each integer by the same factor.
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7.1/7.2 Homework
Determine if the given side lengths belong to an acute, right, or obtuse triangle.
1) 4, 6, 9
2) 4.5, 6, 7.5
3) 5, √ , 8
For each of the following, (1) draw a diagram and (2) solve the problem.
Write answers in decimal form and in simplest radical form.
4) A 12 foot rope is fastened to the top of a flagpole. The rope reaches a point on the
ground 6 feet from the base of the flagpole. What is the height of the flagpole?
5) Arlo and Janie biked 12 miles directly east and then 6 miles directly north. How far
are they from their starting point?
6) Square ABCD has a perimeter of 25 centimeters. What is the length of its diagonal ̅̅̅̅?
7) Suppose that the radius of Earth is approximately 3960 miles. If a
satellite is orbiting at about 7 miles above the Earth, what is the
approximate length of x? Note: The figure is not drawn to scale.
x
7 mi
Earth
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8) If C is the center of the circle and
AB
√ cm, then what is the
length of the circle’s radius?
9) Use the Pythagorean Theorem to find the
distance between point A and point B.
Write your answer in decimal and radical
form.
10) You have a garden in the shape of a right triangle with the dimensions shown.
a) Find the perimeter of the garden.
b) You are going to plant a post every 15 inches around
the garden. How many posts do you need?
c) You plan to attach fencing to the posts around the garden. If each post costs
$1.25 and each foot of fencing costs $0.70, how much will it cost to enclose the
garden?
11) Kim walked diagonally across a rectangular field that measured 100 ft. by 240 ft.
Which expression could be used to determine how far Kim walked?
A √
B √
C
D √
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√
7.3 NOTES
Similarity in Right Triangles
Theorem 7.5
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles
formed are similar to the original triangle and to each other.
Term
GEOMETRIC MEAN
Definition
The geometric mean of two positive numbers a and b is the
positive number x that satisfies
So
and
√
.
.
Example 1:
Find the geometric mean of 6 and 12.
Example 2:
Find the geometric mean of and 25.
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Parachute Man
Scenario #1𝑥
x
𝑎 𝑐
or
𝑥
𝑎
𝑎
𝑏
b
a
-------------------- c -------------------------
Scenario #2𝑦
y
𝑏 𝑐
or
𝑦
𝑏
𝑏
b
a
--------------------- c ------------------------
Scenario #3-
ℎ
h
b
21
a
𝑏 𝑎
𝑎
Examples:
1) Solve for x.
x
6
-------------------- 14 -----------------------
2) Solve for y.
3) Solve for y.
y
4
5
22
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Find the geometric mean between each pair.
1) 24 and 48
2) 16 and 18
3) 4 and 25
4) 6 and 20
Find the value(s) of the variables.
5)
6)
7)
8)
9)
10)
7.3 HOMEWORK
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7.4 (Day 1) NOTES
Special Right Triangles (45°-45°-90°)
Discovery Activity
1) Here are 5 different squares. For each square, draw a diagonal and use the
Pythagorean Theorem to find the length of each diagonal. Write the length of the
diagonal in simplified radical form.
Length of side
Length of Diagonal
(simplified radical form)
Work (Pythagorean Theorem)
1 unit
2 units
3 units
4 units
5 units
2) According to the data in the table, what should be the length of the diagonal of a
20-unit by 20-unit square? ___________
How long is the diagonal of a square with sides that are 65 units? _________
Summary: In a _______°-_______°-_______° ____________________________ triangle, the
length of the hypotenuse is ___________ times the length of one leg.
***The ratio of the sides is _______ - _______ - _______.
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EXAMPLES: (Write all answers in the simplest radical form.)
1) The following triangle was formed
by cutting a square along its
diagonal. What is the length of the
hypotenuse?
3) Find the length of a leg.
4) Calculate the length of the hypotenuse.
5) Calculate the length of the hypotenuse.
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2) The following triangle was formed
by cutting a square along its
diagonal. What is the length of the
leg (the side of the square)?
7.4 (Day 1) HOMEWORK
Find the unknown side measures. Write answers in simplest radical form.
1)
y
2)
3)
y
x
3
4)
5)
8
2
x
7)
6)
y
x
8
y
8)
9)
14
x
y
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10) The blades of a helicopter meet at right angles and are all the same length. The
distance between the tips of two consecutive blades is 36 ft. Find the length of one of
the blades.
11) A ladder is leaning against a house. The base of the ladder is 4 ft from the house and
makes a 45° angle with the ground. How long is the ladder?
12) An isosceles right triangle has legs of length √ . What is the
length of the hypotenuse? Draw a diagram and label the measures.
Record your answer and fill in the bubbles on the grid.
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7.4 (Day 2) NOTES
Special Right Triangles (30°-60°-90°)
Discovery Activity
1) Here is an equilateral triangle. What are the angle measures? If you draw an altitude,
what happens?
2) Here are 3 equilateral triangles. For each triangle, draw an altitude and the resulting
right triangle. Write the length of the short leg, and then use the Pythagorean Theorem
to find the length of the longer leg. Write this length as a simplified radical.
3)
Length of side (hypotenuse)
Length of
shorter leg
Length of longer leg
(Pythagorean Theorem, simplified radical)
Hypotenuse = 2 units
Hypotenuse = 8 units
Hypotenuse = 20 units
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4) What pattern do you notice? According to the data in the table, what should be the
lengths of the legs of a 30°-60°-90° triangle with a hypotenuse of 100 units?
Short leg _________, long leg _________
Summary: In a _______°-_______°-_______° ____________________________ triangle, the
length of the hypotenuse is _______ times the length of the short leg,
and the long leg is ________ times the length of the short leg.
***The ratio of the sides is _______ - _______ - _______.
Note: The ___________ leg is opposite the _____° angle,
and the ___________ leg is opposite the ______° angle.
Examples:
1) The following right triangle is half of an equilateral triangle. What are the lengths of the
short leg and long leg?
2) Given the length of the short leg in the following triangle is 11 cm, what are the lengths
of the long leg and hypotenuse?
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For each of the following, find the two missing side lengths. No decimals!
3)
4)
5)
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7.4 (Day 2) HOMEWORK
Find the missing side lengths. Write answers in simplest radical form.
1)
2)
3)
4)
5)
10
x
y
6)
y
y
x
7)
8)
x
9)
13
60
17
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Problem Solving. Make a sketch and label appropriately.
10)A ladder leaning against a house makes a 60  angle with the ground. If the ladder is 12
feet long, how far from the house is the base of the ladder?
11) An equilateral triangle has a perimeter of 18 cm. How long is the altitude?
12)Each half of the drawbridge is about 284 feet
long, as shown below. How high does a seagull
(who is on the end of the drawbridge) rise when
the angle with measure x° is 30°? 45°? 60°?
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The Story of SOHCAHTOA
Once upon a time, a long time ago, there lived a poor girl on a farm. Her name was Wilma, she
lived alone. She lived a humble life that didn’t involve too much change. She started each day
tending her onion and carrot garden and ended each day with her favorite passion. She loved
dancing. Every night, as the sun began to set, she would retire into her cottage, turn on her CD,
and dance the night away.
Meanwhile, up at the castle, Prince Fred was wondering what the rest of the world was like. He
hopped on his handsome steed and set out to find out what others in the Kingdom did with their
days. He made his rounds and bumped into a million different people, but nothing made much of
an impression with him until he heard a sound. He rode over to Wilma’s cottage and listened to
the unfamiliar rhythm he heard. He knocked on the door and asked what she was doing. She
explained that she was dancing to music. He asked her if she would be willing to teach him how to
dance and she accepted immediately.
Fred was a very slow learner when it came to dancing. It appeared that no matter how hard
Wilma tried to get him into the right step, he would somehow loose the beat and wind up stepping
on her toe. He left at midnight, but asked if she would mind if he came back for another lesson.
Needless to say, the dancing lessons went on, and they fell in love. Fred soon proposed marriage
to Wilma and she, of course, accepted. King Fred and Queen Wilma continued their dancing hobby
every Friday night. The only problem for Queen Wilma was that no matter how hard she tried to
teach King Fred, he remained terribly clumsy. And whenever they went dancing, he would always
step on Queen Wilma’s big toe.
Queen Wilma learned to live with this slight flaw in their true love relationship. But this big-toe
stomping was no laughing matter. So each Saturday morning, after one of their big dancing dates,
Wilma would wake up early and sneak out of the castle to her favorite pond. There she would sit
for an hour, soaking her big toe until the swelling went down.
As the years passed by, the whole kingdom would watch the King and the Queen’s ritual of
dancing and soaking with delight. Even after the King and the Queen died, their love story lived
on in the hearts of the kingdom. Of course, like all good legends, time changed the facts, as well as
the names. King Fred was remembered as King Klutz and Queen Wilma was remembered as
Queen Sohcahtoa (soak a toe a).
You can determine the height of objects using TRIGNOMETRIC RATIOS.
A TRIGNOMETRIC RATIO is a ratio of the lengths of two sides of a right triangle.
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37
Trigonometric Rations (Finding Sides)
7.5/7.7 (Day 1) NOTES
A trigonometric ratio is the ratio of the lengths of two sides in a right triangle. You will
use these to find either missing side lengths or the acute angle measures in a right
triangle.
The three trig ratios are: sine, cosine, and tangent.
Given
below with acute angle A,
B
The opposite leg
is the one across
from the angle.
opposite
leg
C
The adjacent leg is the
one touching the angle.
hypotenuse
adjacent leg
A
opposite leg to A
=
hypotenuse
adjacent leg to A
cos A =
=
hypotenuse
Sine
SOH
sin A =
Cosine
Tangent
tan A =
CAH
opposite leg to A
=
adjacent leg to A
TOA
Use the diagrams and your calculator to complete the chart below. Label the diagrams
with respect to angle A. Your calculator must be in “degree mode”.
Trigonometric
Ratio
Abbreviation Definition
sine A
sin A
cosine A
cos A
tangent A
tan A
Ratio
from
Picture
Decimal
opp  A
hyp
adj A
hyp
opp  A
adj A
A
13
12
B
C
5
Example 1: Find the tan K and the cos J as both a fraction and decimal.
tan K = ________________________________________
cos J = ________________________________________
38
Example 2: Find the values of x and y.
(a)
y
(b)
Steps:
1. Label the sides with respect to the
given angle.
2. Use the appropriate trig ratio to
write a proportion.
3. Solve the proportion for x.
Something to remember:
Sines, cosines, and
tangents are just
NUMBERS! So you can
multiply and divide with
Angle of Elevation/Depression
When you look up at an object, the angle your
line of sight makes with a horizontal line is
called the angle of elevation.
If you look down at an object, the angle your
line of sight makes with a horizontal line is
called the angle of depression.
Example 3:
Leo is sitting in a seat on top of a 200-foot Ferris wheel looking down at his brother Jason
on the ground at an 80° angle of depression. How far is Jason from the base of the Ferris
wheel? Hint: Draw and label a diagram.
39
7.5/7.7 (Day 1) HOMEWORK
Find the indicated trigonometric ratio as a fraction and as a decimal rounded to
three decimal places.
M
1) sin M
2) cos L
3) tan M
30
K
34
16
L
Use your calculator to find the value of each ratio to three decimal places.
4) sin 12  = ____________
5) cos 32  = _______________
Use a trig function to write an equation for each x and y. Then solve.
6)
7)
8
x
36
36
9)
8)
y
5 50
9
10)
7.2
70
y
K
x
J
x
64
15
11)
L
40
12) A handicap ramp has a vertical rise of 24 inches. If the ramp makes an angle of 8 
with the ground, how long is the ramp in feet? Draw a picture and label it. Write an
equation and solve it.
13) Use the diagram to find the distance across the suspension bridge.
14) The distance from the point directly below a kite to the point where the kite is
anchored to the ground with a string is 84 feet. The angle of elevation along the string
to the kite is 65  . Determine the height of the kite, to the nearest tenth of a foot.
15) A lookout tower is 43 m tall. The angle of depression from the top of the tower to a
forest fire is 5  . How far away from the base of the tower is the fire?
16) A flagpole casts a shadow 4.6 m long. The angle of elevation of the sun is 49  . How
tall is the flagpole?
41
7.5/7.7 (Day 2) NOTES
Trigonometric Functions (Finding Angles)
The INVERSE trigonometric functions are used to find ANGLE MEASURES from given
sides of a right triangle.
The table below shows the format for setting up equations with inverse trigonometric
functions to find angle measures.
Given
below with acute angle A,
B
The opposite leg
is the one across
from the angle.
opposite
leg
C
A
adjacent leg
To Find SIDE Measures
opposite
Sine
sin A =
hypotenuse
adjacent
Cosine
cos A =
hypotenuse
Tangent
The adjacent leg is the
one touching the angle.
hypotenuse
opposite
tan A =
adjacent
To Find ANGLE Measures
Inverse Sine
 opposite 
mA  sin1 

 hypotenuse 
Inverse Cosine
 adjacent 
mA  cos1 

 hypotenuse 
Inverse
Tangent
 opposite 
mA  tan1 

 adjacent 
Examples: Find the value of x.
1)
2)
3)
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SOLVING A RIGHT TRIANGLE
To solve a right triangle means to find the measures of ALL its missing sides and angles.
You can solve a right triangle if you know either of the following:
1. Two side measures
2. The measures of one side and one acute angle
If using the Pythagorean Theorem or special right triangle ratios is not possible based on
the given information, you use SIN, COS, or TAN to calculate side measures. You use SIN-1
(inverse SIN), COS-1 (inverse COS), or TAN-1 (inverse TAN) to calculate angle measures.
Example 1: Solve the right triangle (given two side measures).
Example 2: Solve the right triangle (given one side and one angle).
43
7.5/7.7 (Day 2) HOMEWORK
Use the diagram to find the indicated measurement. Round to the nearest tenth.
1)
2)
3)
Use the inverse trig functions to find the indicated angle measure in each right triangle.
4)
5)
6)
7)
8)
9)
10) Solve the triangle.
44
For each problem, make a diagram (if needed), and label the sides. Write a
proportion using the most appropriate trig ratio and solve the proportion. Be sure
to answer the question.
11) A clinometer is an instrument used to measure angles of
elevation, slope, or incline. Sam uses a homemade
version to estimate the height of a tall tree. He stands
about 60 feet away from the tree and looks through the
clinometer and records the angle of elevation. After
calculating using trig functions, he determines the tree
to be about 91 feet tall.
What angle did he record?
12) Bobby wants to set his viewscope so that he
can see the tops of the mountains. The
mountains are known to be about 3000 feet
in elevation. He stands at a point known to
be about a mile (5280 feet) away ,
At what angle should he set his viewscope?
13) A lighthouse is built on the edge of a cliff near
the ocean, as shown in the accompanying
diagram. From a boat located 200 feet from
the base of the cliff, the angle of elevation to
the top of the cliff is 18° and the angle of
elevation to the top of the lighthouse is 28°.
What is the height of the lighthouse, x, to the
nearest tenth of a foot?
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UNIT 7 PERFORMANCE TASK
Trigonometric Ratios
Each student will create an original word problem that demonstrates using a trig triangle
in a real world situation. This project should be done on a blank sheet of paper. It should
show creativity and use of colors. Pictures may be hand drawn or may be taken from a
magazine or other periodical. Ink, crayons, color pencil are accepted (regular pencil is
not). This project is worth a quiz grade.
Example:
Miss Bauer went for a bicycle ride in Chicago. She biked 115 feet to the middle of the
bridge where she stopped to take a picture. After she rode off the bridge, the bridge
opened up to let sailboats go through. The bridge opened up 50 degrees. How high did the
bridge open?
sin = Opposite
Hypotenuse
sin
sin
ft.
46
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