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Geometry Rules! Chapter 8 Notes
Notes #6: The Pythagorean Theorem (Sections 8.2, 8.3)
A. The Pythagorean Theorem
Right Triangles: Triangles with _____________ right angle
Hypotenuse: the side across from the _________ angle
(the _______________ side of the triangle)
Legs: the sides across from the __________ angles (the two
____________________ sides of the triangle)
Pythagorean Theorem
(leg)2 + (leg)2 = (hypotenuse)2
Solve for x:
 Find and label c across from the right angle
 Label a and b
 Write and solve a2 + b2 = c2
1.)
2.)
6
13
3
x
x
12
3.)
4.)
5
5
3
x
2 3
x
2x
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5.)
6.) A rectangle has length of 4cm and a
width of 2cm. Find the length of its
diagonal. (Hint: draw a picture first!!)
12
10
x
x
10
24
7.) The perimeter of a square is 20in. Find
the length of its diagonal.
8.) The diagonals of a rhombus have
lengths of 6ft and 8ft. Find the perimeter of
the rhombus. (Hint: what do we know
about the diagonals of a rhombus?)
B. Classifying Triangles
We can also use the Pythagorean Theorem to classify a triangle
Acute:
Right:
Obtuse:
a 2  b2  c2
a 2  b2  c2
a 2  b2  c 2
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Classify the triangle with the given sides as acute, right, or obtuse. If the triangle is not
possible, say so:
 Check that the triangle is possible (short side + short side > long)
 Compare a2 + b2 vs. c2
9.) 6, 8, 10
10.) 5, 6, 7
12.) 2, 5, 6
11.) 2, 4, 6
C. Algebra Practice: Solving Quadratics
 Get all terms to one side and equal to zero. Arrange in descending order
 Factor completely
 Set each ( ) = 0; solve each equation
Solve
13.) 2x2 – 6x = 0
16.) 3x2 + 5x = 2
14.) x2 – 3x = 10
15.) x2 = 36
17.) 4x2 – 8x +2 = 3x + 5
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Notes #7: Geometric Means and Similar Right Triangles (Section 8.1)
A. Geometric Mean
asks the question: “what number, squared, equals the product of two given numbers?”
Find the geometric mean of the listed numbers:
 Use the given numbers in this equation: x2 = ab
 Solve for x
1.) 9 and 16
2.) 12 and 3
3.) 5 and 15
B. Similar Right Triangles
 When an altitude of a right triangle is drawn to its hypotenuse, three similar right
triangles are formed:
y
x
a
z
b
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Solve for the variables:
 Re-draw the three triangles and label all sides
 Set up proportions to solve for the variables
 Look for ways to use the Pythagorean theorem
4.)
p
n
m
5
20
5.)
1
4
a
1
9
b
c
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6.)
3
5
y
x
z
C. Algebra Practice
Solve for x by factoring:
7.) x3 – 2x2 = -18 + 9x
8.) 2x3 – 4x2 = 16x
9.) 3x3 = 48x
10.) 2(2x2 – 5x) = 9 – 7x – 2x2
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Notes #8: Special Right Triangles (Section 8.4)
A. 45 ◦– 45 ◦ – 90 ◦ Triangles
 Solve for the missing sides using ITT and Pythagorean Theorem
 Write a rule based on the pattern
1.)
2.)
3.)
8
45
45
45
5 2
45
45
4
45 ◦– 45 ◦ – 90 ◦ Triangles
45
45
Solve for x:
 Find the side for which you have a value
 Set this side = to its rule
 Solve for n
 Plug n back into the triangle rules find the length of all sides
4.)
5.)
45
45
6
2
45
45
3
45
-86.)
7.)
45
2
45
2
8
45
45
8.)
9.)
45
2
3
7
45
B. 30 ◦– 60 ◦ – 90 ◦ Triangles
 Look for the pattern in the triangles below
a)
b)
60
60
10
4
5
2
30
2
3
30 ◦– 60 ◦ – 90 ◦ Triangles
30
5
3
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Solve for the missing sides:
 Find the side for which you have a value
 Set this side = to its rule
 Solve for n
 Plug n back into the triangle rules to find the length of all sides
10.)
11.)
60
30
3
4
30
3
60
12.)
13.)
60
5
12
3
60
30
30
14.)
15.)
60
60
9
30
30
6
16.)
17.)
30
x
8
30
60
y
60
10
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Notes #9: Right Triangle Trigonometry (Sections 8.4, 8.5, 8.6)
Solve for the variables:
1.)
2.)
45
x
y
y
x
45
45
20
45
3 6
3.)
4.)
30
8 5
y
9 2
60
30
60
x
5)
6)
x
z
40
y
3
z
6
y
x
2
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B. 3 Trig Functions
Trigonometry relates a right triangle’s ________ to its ___________ ___________.
Sine(angle) = _____(angle) =
Cosine(angle) = ______ (angle) =
Tangent(angle) = ______(angle) =
OR
______(angle) =
Complete the triangle and find each value as a simplified fraction:
7.) sinA
8.) sinB
B
10
9.) cosB
4
10.) cosA
A
11.) tanA
C
12.) tanB
C. Using your Trig Table
 If you know the angle, find the angle in the left-most column and read to the right to
find its sine, cosine, and/or tangent
 If you know the decimal value of its sine, cosine, and/or tangent, look down the
sin/cos/tan column until you find the closest match. The read to the left to find the
angle.
Use your trig table. Round your answers to the nearest hundredth:
13.) a) sin32◦ = _____
b) tan19◦ = _____
c) cos75◦ = _____
d) cos48◦ = _____
e) sin80◦ = _____
f) tan59◦ = _____
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Use your trig table. Round your answers to the nearest degree:
14.) a) sin____◦ = 0.9903
b) tan____◦ = 0.21
c) cos____◦ = 0.79
d) cos____◦ = 0.454
e) sin____◦ = 0.7
f) tan____◦ = 2.5
D. Using your Calculator (make sure your calculator is in degree mode!!)
 If you know the angle, type the sin/cos/tan button, then the angle, then enter/equals.
It should look like this on your calculator screen:
sin(37) = 0.601815…
 If you know the decimal value of its sin/cos/tan, type the 2nd/Shift Key, then the
decimal, then enter/equals. It should look like this on your calculator screen:
cos-1(0.5) = 60
Use your calculator. Round your answers to the nearest hundredth:
15.) a) sin32◦ = _____
b) tan19◦ = _____
c) cos75◦ = _____
d) cos48◦ = _____
e) sin80◦ = _____
f) tan59◦ = _____
Use your calculator. Round your answers to the nearest degree:
16.) a) sin____◦ = 0.9903
b) tan____◦ = 0.21
c) cos____◦ = 0.79
d) cos____◦ = 0.454
e) sin____◦ = 0.7
f) tan____◦ = 2.5
E. Solving Quadratics using the Quadratic Formula
 Not all quadratics (ax2 + bx + c = 0) can be factored
b  b2  4ac
 In this case, use the quadratic formula: x 
2a
 Reduce the radical expression and reduce the fraction, if possible
17.) 7x2 + 10x + 3 = 0
18.) x2 – 6x + 3 = 0
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19.) x – 3x = – 1
2
20.) 4x + 20x = – 25
2
Notes #10: Right Triangle Trigonometry (Sections 8.5, 8.6, 8.7)
A. Constructing Right Triangles
Complete without your calculator:
3
5
2
, find sinB.
3
1.) If sin A  , find tan A .
2.) If cos B 
5
3.) If sin A  , find tan2A
8
4.) Find the altitude of an equilateral
triangle with perimeter 12 in.
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B. Solving for Missing Sides of Right Triangles
 Pick an acute angle; label sides as O(opposite), A(adjacent), and H(hypotenuse)
 Choose a trig function (sine, cosine, tangent), write an equation
 Solve for the variable – wait to use your calculator until the last step
Write an equation to solve for each variable. Round sides to the nearest tenth and angles
to the nearest whole degree:
5.)
z
y
x
38
46
6.)
x
y
z
55
48
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C. Solving for Missing Angles of Right Triangles
 Follow the same steps, but remember that when you are solving for an angle, you
must use the 2nd/Shift Key (sin-1) and/or use your trig table from right to left.
Find missing sides in reduced radical form and find missing angles to the nearest whole
degree:
7.)
5
y
2
z
x
8.)
y
10
z
x
6
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Notes #11: Applications of Right Triangle Trigonometry
A. Angles of Depression and Elevation (all relative to a horizontal line of sight)
Solve each problem:
 Draw a picture, include:
line of sight
angle of depression/elevation
labeled right triangle
 Write an equation and solve
Solve for the missing information:
1.) The angle of elevation of a ramp is 25˚.
If the ramp is 6m off the ground at its
highest point, how long is the inclined
surface of the ramp?
2.) A streetlight casts a 5ft shadow. If the
streetlight is 9ft tall, what is the angle of
elevation of the sun from the ground?
3.) Joe is out flying his kite. He has let out 100 ft of string and knows that the kite is 65ft
off the ground. What is the angle of elevation of the kite string from the ground?
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Classwork #11: Chapter 8 Review
Special Right Triangles: Solve for x and y in reduced radical form.
1.)
2.)
45
30
y
6 3
y
21
45
x
60
x
3.)
2
4.)
45
3
5
60
30
y
45
x
Similar Right Triangles: Solve for m, n, and p in reduced radical form.
5.)
p
n
m
5
10
Word Problems: Leave answers in reduced radical form (no decimals!)
6.) The altitude of an equilateral
triangle is 6ft. Find its perimeter.
7.) The perimeter of a square is 24m.
Find the length of its diagonal.
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8.) The hypotenuse of a 45˚, 45˚, 90˚
triangle is 10in. What is the length of
one of its legs?
9.) The diagonals of a rhombus are
10cm and 24cm long. What is the
perimeter of the rhombus?
Using Trigonometry: Solve for the indicated quantity. Round lengths to the nearest
tenth and angles to the nearest whole degree.
10.)
11.)
8
y
42
x
55
11
12.)
13.) A flagpole casts a shadow that is
30ft long. If the angle of elevation to the
sun is 31 degrees, how tall is the
flagpole?
10
x
4
14.) A 12ft ladder leans against a building in such a way that its base is 4ft from the
building. What is the angle of elevation of the ladder to the building?
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Notes #12: Circles and Key Vocabulary (Section 9.1)
1.) Circle O (written as _____ O) has center ___
B
O
A
C
2.) OA is a __________ of the circle. This is a segment
connecting the ___________ to any ___________ on the
circle. Other radii: _____, _____
3.) AB is a _______________ of the circle. This is a
segment connecting two _____________ on the circle and
passing through the circle’s ______________.
4.) What is the relationship between a radius and a
diameter?
_______________________________________________
5.) AB and MN are __________. These
segments connect any ____ points on a
circle.
N
A
M
6.) What is a name for the longest chord
in a circle? ___________
7.) AB and MN are ____________.
These are lines that contain a _________.
B
8.) A ______________ is a segment, ray, or line that
touches a circle only once. Name four tangents: ______
______, ______, ______
9.) The point where a tangent touches a circle is called the
__________ _____ ________________. Name the point of
tangency: ____
O
Z
Y
X
10.) A tangent is always ___________________ to the
radius at the point of tangency. _______  _______
11.) Name two right angles: _________, ________
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12.) Circles and spheres are called _____________________ if they have the same
center.
13.) Draw two concentric circles and two concentric spheres
concentric circles
concentric spheres
14.) We say that a polygon is _________________ ______ a circle when all vertices
(corners) are on the circle.
In this case, we can also say that the circle is _______________________ ________
about the polygon.
15.) Describe this figure in two ways:
B
C
A
P
D
16.) Describe this figure in two ways:
G
Q
F
H
17.) Draw a triangle inscribed
in a circle.
18.) Draw a circle circumscribed about
a rectangle.
- 21 Geometry Chapter 8 Study Guide: Right Triangle Geometry
Radical Expressions: Simplify each expression
2 3
1. 3 24
2.
2
3 5 
3.
2
4.
Pythagorean Theorem: Solve for x
5.
6.
 4 6 3 2 
7.
5
x
3 3
8
x
x
8
12
3
11
8. Find the length of a diagonal of a square with
perimeter 20m.
9. The diagonals of a rhombus have length 8cm and
6cm. Find the perimeter of the rhombus.
State whether the triangle with the given sides is not possible, right, acute, or obtuse:
10. 4, 6, 8
11. 1, 4, 6
12. 8, 10, 12
Geometric Means and Similar Triangles:
13. Find the geometric mean of 5 and 10
14. Find the geometric mean of 4 and 20.
15. Solve for x, y, and z: (hint: use 3 triangles)
16. Solve for x, y, and z: (hint: use 3 triangles)
x
4
z
y
25
y
x
z
4
16
- 22 Special Right Triangles: Solve for x and y
17.
18.
60
y
19.
30
4
5 3
30
60
30
x
x
20.
x
6
y
y
60
21.
22.
45
y
x
y
30
x
45
45
y
9
8 2
45
7
60
x
23.
24.
25.
y
45
10
x
60
45
45
x
3 5
y
2 2
x
45
30
y
Right Triangle Trigonometry:
26. Find sinA, cosA, and tanA as fractions
27. Find sinB, cosB, and tanB as fractions
B
C
6
6
2
A
B
10
A
C
Solve for x and y; round to the nearest tenth or leave in radical form:
28.
29.
x
10
26
y
y
9
51
x
- 23 30.
31.
4
x
y
10
3
x
y
4
Applications of Trigonometry:
32. If the sun’s angle of elevation is 48˚ and a flag
pole casts a shadow that is 40 feet long, how tall is
the flag pole?
34. CCA students want to design a water park and
have a particular slide in mind. They want it to be
perfectly straight and 40ft high, and for students to
have a 60ft long slide-ride. What must the angle of
elevation of the slide be? (Round to the nearest
whole degree.)
33. From a lighthouse that is 150m above the shore,
the angle of depression to a ship is 20˚. How far is
the ship from the shore?
35. How far from the base of a building is the
bottom of a 30ft ladder that makes an angle of 75˚
with the ground?
Solve by Factoring:
36. x2 + 5x – 6 = 0
37. 2x2 + 7x = 0
38. 5x2 = 80
39. x2 – 3x = 18
40. x3 – 4x2 = 45x
41. x3 + x2 = 16x +16
42. 15 + 4x2 = 17x
43. x3 – 3x2 = 4x - 12
Solve by using the quadratic formula:
44. 3x2 + 3x = 4
45. x2 + 5x + 2 = 0
46. x2 + 2x – 1 = 0
47. x(x + 5) = 14
48. x(x + 6) = -4
49. x2 = -6x - 2