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Geometry B
Ms. Hanning
Name: ______________________________________________ Hour: _________
Test Date: ____________________ Grade: _______/70 participation points
Chapter 7 Note Packet
Right Triangles
and
Trigonometry
To receive full credit on your notes, all notes must be completed and turned in on test day.
2
3
Goals
Find side lengths in right triangles.
One of the most famous theorems in mathematics is the Pythagorean Theorem, named for the
ancient Greek mathematician Pythagoras (around 500 B.C.). This theorem can be used to find
information about the lengths of the sides of a right triangle.
Find the length of the hypotenuse of the right triangle.
A 16 foot ladder rests against the side of the house, and the base of the ladder is 4 feet
away. Approximately how high above the ground is the top of the ladder?
Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle.
Write your answer in simplest radical form.
1. 2.
3. A 5 foot board rests under a doorknob and the base of the board is 3.5 feet away from the bottom of the
door. Approximately how high above the ground is the doorknob?
Find the length of a hypotenuse Example 1
Find the length of a leg Example 2
Check Point
Section 7.1 – Apply the Pythagorean Theorem
Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of
the squares of the lengths of the legs.
4
Common Pythagorean Triples and some of Their Multiples
3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
6, 8, 10 10, 24, 26 16, 30, 34 14, 48, 50
9, 12, 15 15, 36, 39 24, 45, 51 21, 72, 75
30, 40, 50 50, 120, 130 80, 150, 170 70, 240, 250
3𝑥, 4𝑥, 5𝑥 5𝑥, 12𝑥, 13𝑥 8𝑥, 15𝑥, 17𝑥 7𝑥, 24𝑥, 25𝑥
The most common Pythagorean triples are in bold. The other triples are the result of
multiplying each integer in bold face triple by the same factor.
Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters.
Pythagorean Triples
A Pythagorean triple is set of three positive integers a, b, and c that satisfy the equation 𝑐2 = 𝑎2 + 𝑏2 .
Find the length of the hypotenuse of the right triangle.
Find the area of the triangle.
4. 5.
Find the unknown side length of the right triangle using the Pythagorean Theorem. Then use a
Pythagorean triple.
6. 7.
Find the length of a hypotenuse using two methods Example 4
Check Point
Find the area of an isosceles triangle Example 3
5
Goals: Find possible side lengths of a triangle.
Below is a drawing an obtuse scalene triangle. Notice that the longest side and the largest angle are opposite
each other and the shortest side and the smallest angle are opposite each other. These relationships are true for
all triangles are stated in the theorems below. These relationships can help you decide whether a particular
arrangement of side lengths and angle measures in a triangle may be possible.
You are constructing a stage prop that shows a large triangular mountain. The bottom edge of the mountain is
about 27 feet long, the left slope is about 24 feet long, and the right slope is about 20 feet long. You are told that
one of the angles is about 46° and one is about 59°. What is the angle measure of the peak of the mountain?
a) 46°
b) 59°
c) 75°
d) 85°
Pre-Requisite: Inequalities in Triangles
Standardized Test Practice Example 1
Theorems
Theorem 5.10
If one side of a triangle is longer than another side, then the angle
opposite the longer side is larger than the angle opposite the shorter side.
Theorem 5.11
If one angle of a triangle is larger than another angle, then the side
opposite the larger angle is longer than the side opposite the smaller angle.
Theorem 5.12
The sum of the lengths of any two sides of a triangle is greater than the
length of the third side.
6
A triangle has one side length 12 and another of length 8. Describe the possible lengths of the third side.
List the sides and the angles in order from smallest to greatest.
1. 2.
Is it possible to construct a triangle with the given side lengths? If not, explain why not.
3. 6, 7, 11 4. 3, 6, 9
5. Describe the possible lengths of third side of the triangle with side lengths 3 meters and 4 meters.
Find the possible side lengths Example 2
Check Point
7
Goals: Use the Converse of the Pythagorean Theorem to determine if a triangle is a right triangle.
The converse of the Pythagorean Theorem is also true. You can use it to verify that a triangle with given side
lengths is a right triangle.
Tell whether the given triangle is a right triangle.
a) b)
Tell whether a triangle with the given side lengths is a right triangle.
1. 4, 4√3, and 8 2. 10, 11, and 14 3. 5, 6, and √61
a) b)
Section 7.2 – Use the Converse of the Pythagorean Theorem
Verify right triangles Example 1
Theorem 7.2: Converse of the Pythagorean Theorem
If the square of the length of the longest side of a triangle is equal to the sum of the
squares of the lengths of the other two sides, then the triangle is a right triangle.
If 𝑐2 = 𝑎2 + 𝑏2, then ∆𝐴𝐵𝐶 is a right triangle.
Check Point
Use Pythagorean Theorem on an Acute & Obtuse Triangle Example 2
4 in 7 in
6 in
5 mi
4 mi
8 mi 72 ? 42 + 62 82 ? 42 + 52
8
Classifying Triangles
The Converse of the Pythagorean Theorem is used to verify that a given triangle is a right triangle. The
theorems summarized below are used to verify that a given triangle is acute or obtuse.
Can segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute,
right, or obtuse?
Tell whether the triangle is a right triangle. If it is not, classify it as acute, obtuse, or equiangular.
4. 5.
Decide whether the numbers can represent the side lengths of a triangle. If they can, classify the triangle
as acute, right, or obtuse.
6. 16, 30, and 34 7. 10, 12, and 30 8. 18, 34, and 45
Classify triangles Example 3
Step 1: Use the Triangle Inequality
Theorem to check that the segments an
make a triangle.
Step 2: Classify the triangle by
comparing the square of the length of
the longest side with the sum of the
squares of the lengths of the shorter
sides.
Check Point
Methods for Classifying a Triangle by Angles Using its Side Lengths
Theorem 7.2 Theorem 7.3 Theorem 7.4
If 𝑐2 = 𝑎2 + 𝑏2, then,
𝑚∠𝐶 = 90° and ∆𝐴𝐵𝐶 is a
________________ triangle.
If 𝑐2 < 𝑎2 + 𝑏2, then,
𝑚∠𝐶 < 90° and ∆𝐴𝐵𝐶 is an
________________ triangle.
If 𝑐2 > 𝑎2 + 𝑏2, then,
𝑚∠𝐶 > 90° and ∆𝐴𝐵𝐶 is an
________________ triangle.
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Theorem 7.8: 𝟒𝟓°- 𝟒𝟓°- 𝟗𝟎° Triangle Theorem
In a 45°- 45°- 90° triangle, the hypotenuse is ______ times as long as each leg.
hypotenuse = leg ______
The extended ratio of the side lengths of a 45°- 45°- 90° triangle is ____ : ____ : ____
Pro
of
for
the
𝟒𝟓
°- 𝟒𝟓
°- 𝟗𝟎
° ∆
Goals: Use the relationships among the sides in special right triangles.
𝟒𝟓°- 𝟒𝟓°- 𝟗𝟎° Triangle.
Let’s see what the relationship is between the side lengths of a 45°- 45°- 90° Triangle. Draw a square below.
Find the length of the hypotenuse.
a) b)
Find the lengths of the legs in the triangle.
a) b)
Section 7.4 – Special Right Triangles
Check Point
Find hypotenuse length in a 45°- 45°- 90° triangle Example 1
Find leg lengths in a 45°- 45°- 90° triangle Example 2
10
Pro
of
for
the
𝟑𝟎
°- 𝟔𝟎
°- 𝟗𝟎
° ∆
The body of a dump truck is raised to empty a load of sand. How high is the 14
foot body from the frame when it is tipped upward at a 45° angle?
Find the value of the variable.
1. 2. 3.
4. Approximate the length of the leg of a 45°- 45°- 90° triangle with a hypotenuse length of 6.
𝟑𝟎°- 𝟔𝟎°- 𝟗𝟎° Triangle.
Let’s see what the relationship is between the side lengths of a 30°- 60°- 90° Triangle. Draw an equilateral triangle
below.
Check Point
Find a height Example 3
Theorem 7.9: 𝟑𝟎°- 𝟔𝟎°- 𝟗𝟎° Triangle Theorem
In a 30°- 60°- 90° triangle, the hypotenuse is _________________ as long as the
shorter leg, and the longer leg is _______ times as long as the shorter leg.
hypotenuse = ______ shorter leg longer leg = shorter leg ______
The extended ratio of the side lengths of a 30°- 60°- 90° triangle is ____ : ____ : ____
10
11
The logo on the recycling bin at the right resembles an equilateral triangle with side lengths
of 6 centimeters. What is the approximate height of the logo?
Approximate the values of x and y.
A car is turned off while the windshield wipers are moving. The 24 inch wipers
stop, making a 60° angle with the bottom of the windshield. How far from the
bottom of the windshield are the ends of the wipers?
Find the value of each variable.
5. 6.
7. A kite is attached to a 100 foot string as shown in the diagram. How far above the ground is the kite
when the string forms a 30° angle with the ground?
Check Point
Find a height Example 6
Find the height of an equilateral triangle Example 4
Find lengths in a 30°- 60°- 90° triangle Example 5
12
Goals: Use the tangent ratio for indirect measurement.
A ____________________________________________ is a ratio of the lengths
of two sides in a right triangle. You will use trigonometric ratios to find the
measure of a side or an acute angle in a right triangle.
The ratio of the lengths of the legs in a right triangle is constant for a given angle
measure. This ratio is called the ______________________ of the angle.
Find tan 𝑆 and tan 𝑅. Write each answer as a fraction and as a decimal rounded to four decimal places.
Find tan 𝐽 and tan 𝐾. Round to four decimal places.
1. 2.
Section 7.5 – Apply the Tangent Ratio
Find tangent ratios Example 1
Check Point
Tangent Ratio
tan𝐴 =length of leg opposite ∠𝐴
length of leg adjacent to ∠𝐴=𝐵𝐶
𝐴𝐶
Let ∆𝐴𝐵𝐶 be a right triangle with acute ∠𝐴.
The tangent of ∠𝐴 (written as tan A) is defined as follows.
Remember these abbreviations: tangent → tan, opposite → opp, and adjacent → adj
13
Find the value of x.
Find the height h of the lighthouse to the nearest foot.
Use a special right triangle to find the tangent of a 60° angle.
Find the value of x. Round to the nearest tenth.
3. 4.
5. Find the height of the flagpole to the nearest foot.
Find a leg length
Example 2
Estimate height using tangent
Example 3
Use a special right triangle to find a tangent
Example 4
Check Point
14
Goals: Use the sine and cosine ratios.
The __________________ and _______________________ ratios are trigonometric ratios for acute angles that
involve the lengths of a leg and the hypotenuse of a right triangle.
Find sin 𝑆 and sin 𝑅. Write each answer as a fraction and as a decmal rounded to four places.
Find 𝐬𝐢𝐧 𝑿 and 𝐬𝐢𝐧 𝒀. Write each answer as a fraction and as a decmal. Round to four decimal places, if
necessary.
1. 2.
Find sine ratios
Example 1
Sine and Cosine Ratio
sin𝐴 =length of leg opposite ∠𝐴
length of hypotenuse=𝐵𝐶
𝐴𝐵
cos𝐴 =length of leg adjacent to ∠𝐴
length of hypotenuse=𝐴𝐶
𝐴𝐵
Let ∆𝐴𝐵𝐶 be a right triangle with acute ∠𝐴.
The sine of ∠𝐴 and cosine of ∠𝐴 (written as sin A and cos𝐴)
are defined as follows.
Remember these abbreviations: sine → sin, cosine → cos, hypotenuse → hyp
Section 7.6 – Apply the Sine and Cosine Ratios
Check Point
Check Point
15
Find cos 𝑈 and cos 𝑊. Write each answer as a fraction and as a decimal (rounded to four decimal places).
You walk from one corner of a basketball court to the opposite corner. Write and solve
a proporton using a trignometric ratio to approximate the distance of the walk.
Find 𝐜𝐨𝐬 𝑹 and 𝐜𝐨𝐬 𝑺. Write each answer as a fraction and as a decmal. Round to four decimal places, if
necessary.
3. 4.
5. A rope, staked 20 feet from the base of a building, goes to the roof and forms an
angle of 58° with the ground. To the nearest tenth of a foot, how long is the rope?
Find cosine ratios
Example 2
Use a trigonometric ratio to find a hypotenuse
Example 3
Check Point
S
16
Use a sine or cosine ratio to find the value of a and b.
Angles
If you look up at an object, the angle your line of sight makes with a horizontal line is alled the _____________
__________________________________. If you look down at an object, the angle your line of sight makes
with a horizontal line is call the _______________________________________________.
You are at the top of a roller coaster 100 feet above the ground. The angle of
depression is 44°. About how far do you ride down the hill?
A railroad crossing arm that is 20 feet long is stuck with an angle of elevation of 35°.
Find the lengths of x and y.
Use trig ratios to find the values of the variables
Example 4
Find a hypotenuse using an angle of depression
Example 5
Find a hypotenuse using an angle of elevation
Example 6
17
Use a special triangle to find the sine and cosine of a 60° angle.
6. Use a sine or cosine ratio to find the value of e and f.
7. A pilot is looking at an airport from her plane. The angle of depression is
29°. If the plane is at an altitude of 10,000 feet, approximately how far is
it from the airport?
8. A dog is looking at a squirrel at the top of a tree. The distance between the two
animals is 55 feet and the angle of elevation is 64°. How high is the squirrel and
how far is the dog from the base of the tree?
9. Use a special triangle to find the sine and cosine of a 30° angle.
Use a special triangle to find a sine and cosine
Example 7
Check Point
18
Goals: Use inverse tangent, sine, and cosine ratios.
To solve a right triangle is to __________________________________________________________________
_________________________________________________________________________________________.
Use a calculator to approximate the measure of ∠𝐴 to the nearest tenth of a degree.
Let ∠𝐴 and ∠𝐵 be acute angles in a right triangle. Use a calculator to approximate the measures of ∠𝐴 and ∠𝐵
to the nearest tenth of a degree.
a) sin 𝐴 = 0.87 b) cos 𝐵 = 0.15
a) b) c)
Section 7.7 – Solve Right Triangles
Check Point
Use an inverse tangent to find an angle measure
Example 1
Use an inverse sine and inverse cosine Example 2
Inverse Trigonometric Ratios
Let ∠𝐴 be an acute angle.
Inverse Tangent: If tan𝐴 = 𝑥, then tan−1 𝑥 = 𝑚∠𝐴.
Inverse Sine: If sin𝐴 = 𝑦, then sin−1 𝑦 = 𝑚∠𝐴.
Inverse Cosine: If cos𝐴 = 𝑧, then cos−1 𝑧 = 𝑚∠𝐴.
Find the measure of ∠𝐴 to the nearest tenth of a degree Example 3
19
Suppose your school is building a raked stage. The
stage will be 30 feet long from front to back, with a
total rise of 2 feet. A rake (angle of elevation) of 5° or
less is generally preferred for the safety and comfort of
the actors. Is the raked stage you are building within
the range suggested?
1. Use a calculator to approximate the measure of ∠𝑄 to the nearest tenth of a degree.
Let ∠𝑪 be an acute angle in a right triangle. Use a calculator to approximate the measures of ∠𝑪 to the
nearest tenth of a degree.
2. sin 𝐶 = 0.24 3. cos 𝐶 = 0.37
Find the measure of ∠𝑨 to the nearest tenth of a degree.
4. 5. 6.
7. You lean a ladder against a wall. The base of the ladder is 4 feet from the wall.
What angles 𝜃 does the ladder make with the ground?
Check Point
Solve a real-world problem Example 4
20
Solve the right triangle formed by the water slide shown in the figure. Round
decimal answers to the nearest tenth.
Solve the right triangle. Round decimal answers to the nearest tenth.
8. 9.
10. You are standing 350 feet away from a skyscraper that is 750 feet tall. What is the angle of elevation
from you to the top of the building?
Solve a right triangle Example 5
Check Point