Download Chapter 6 Similarity Exercises for Similarity:

Chapter 6 Similarity
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
Ratio is a comparison between 2 “things”
A proportion is two equal ratios

Similar Polygons
o Corresponding angles are congruent
o Corresponding sides are proportional
Triangle Similarity
o AA angle-angle similarity, SSS side-side-side similarity, SAS side-angle-side similarity
Parallel lines can intersect 2 or more transversals proportionally
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
Exercises for Similarity:
1.______ Assuming the triangle below are similar, find the height of the building.
2. ____ ΔPGT~ΔRST, what is the length of QT?, If PT = 15, what is the
______ length of RT?
3. a)_____ A soccer team played 24 games and won 16 games. a) What is the ratio of the number of wins
b)_____ to loses b) What is the ratio of the number of games played to games won?
For each pair of figures, determine if there is a similarity transformation that maps one figure onto the other.
4. ______
5._______
6. ________
7
7
7
Chapter 7: Quadrilaterals and Polygons

Identify special quadrilaterals and use their properties to solve problems.
o Parallelogram
 Both pairs of opposite sides parallel and congruent ()
 Both pairs of opposite angles congruent, and consecutive angles are supplementary
 Diagonals bisect each other
o Rectangle
 All the properties of a parallelogram
 4 right angles
 Diagonals are congruent
o Rhombus
 All the properties of a parallelogram
 4 congruent sides
 Diagonals are perpendicular, and diagonals bisect opposite angles
Square
 All the properties of a parallelogram AND rectangle AND rhombus!
o Trapezoid
 Exactly 1 pair of opposite sides parallel
 The midsegment is parallel to the bases and is the average (or ½ the sum) of the
bases
o Isosceles Trapezoid
 All the properties of a trapezoid
 Legs are congruent, and base angles are congruent
 Diagonals are congruent
o Kite
 Exactly 2 pairs of consecutive sides congruent
 Exactly 1 pair of opposite angles congruent (the “elbows”)
 Diagonals are perpendicular
Prove a quadrilateral is a special quadrilateral.
o On a coordinate plane, plot the 4 vertices and
 Use slope (rise/run) to prove parallel (same slope) or right angles (opposite
reciprocals)
 Use Pythagorean Theorem or Distance Formula to prove congruent sides or
diagonals
o Parallelogram: By any of the 5 properties above, OR by showing 1 pair of opposite sides
both parallel and congruent
o Rectangle: It is a parallelogram AND… Congruent diagonals OR 1 right angle
o Rhombus: It is a parallelogram AND… 1 pair congruent consecutive sides OR perpendicular
diagonals OR 1 diagonal bisects opposite angles
o Square: It is a parallelogram AND a rectangle AND a rhombus!
 Polygons
o Interior angles of a polygon found by using (n  2)180 ⁰
o The sum of the exterior angle of a polygon = 360⁰
o A REGULAR polygon is both equilateral and equiangular
o

Exercises for Quadrilaterals and Polygons:
Graph quadrilateral ABCD. Then determine the most precise name for each quadrilateral.
Then find the perimeter of each quadrilateral. Leave your answers in simplest radical form.
7. Name __________
Perimeter __________
A(2, 3), B(-4, 3), C(-2, 6), D(1, 6)
8. Name __________
Perimeter __________
A(0, 6), B(3, 3), C(0, -5), D(-3, 3)
9. Name __________
Perimeter __________
E(0, 4), F(3, 0), G(7, 3), H(4, 7)
10. __________ QUVX is a rectangle with Q(-7, -3) and Z(-2, 1). What are the coordinates of U?
11. Parallelogram
x = __________
y = __________
12. Parallelogram
x = __________
13. Rhombus
1 = _____, 2 = _____
perimeter = __________
14._______ Find the sum of the interior angles of a decagon.
15._______If one exterior angle of a regular polygon is 18⁰, how many sides does the polygon have?
Find the values of the unknowns in each figure.
16.______
17.______
18._______
Chapter 8: Right Triangles and Trigonometry
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
Apply
o
o
o
the Pythagorean Theorem and Pythagorean Triples to find unknown sides.
a2 + b2 = c2
or leg2 + leg2 = hyp2
Some triples are 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41
You will often be asked to solve a multi-step problem which requires Pythagorean Theorem
or triples but then asks for the perimeter, area, etc.
Identify and apply special right triangle relationships to find lengths of unknown sides.
30°
1
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
60°
3
90°
2
OR
45°
45°
1
1
90°
2
Simplify radicals, add/subtract/multiply/divide radicals, and rationalize the denominator.
Use trigonometric ratios to find unknown sides or angles of right triangles.
o SOH CAH TOA:
opposite
sin  
hypotenuse
adjacent
cos  
hypotenuse
opposite
tan  
adjacent
To find an unknown side, set up the equation and solve as a proportion.
To find an unknown angle, press 2ND SIN or 2ND COS or 2ND TAN to get the inverse
operation.
Find the angle of elevation or angle of depression, or use them to find unknown sides.
o Angle of elevation/depression is always starting at the HORIZONTAL and angling up or
down.
o An exception is if the problem asks for the angle “from the vertical” or “the angle between
the ladder and the building.”
o
o

angle between
the ladder and
the building
Exercises for Right Triangles and Trigonometry:
Do the given length describe a right triangle?
19. __________ 14, 48, 50
20. __________ 6, 7, 9
Angle of
elevation
Find the values of the variables. Leave your answers in simplest radical form.
21. x = __________
22. x = __________
y = __________
23. x = __________
y = __________
What are the measures of
the two congruent angles?
Find the value of x. Round lengths of segments to the nearest tenth and angle measures to the nearest
degree.
24. __________
25. __________
26. __________ A surveyor measures the top of a building 50 ft. away from him. His angle-measuring
device is 4 ft. above ground. The angle of elevation to the top of the building is 63°. How tall is the
building?
27. __________ A forest ranger looking out from a ranger’s station can see a forest fire at a 35° angle
of depression. The ranger’s position is 100 ft. above the ground. How far is it from the ranger’s station
to the fire?
28. __________ A moving van traveled 200 mi west, then 70 mi south, then 50 mi east, and finally
100 mi north. Find the distance from the point of origin to the destination (to the nearest mile), and
the angle from the origin to the destination (to the nearest degree).
Find the value of each variable. Leave your answers in simplest radical form. Then find the perimeter
and area of each composite figure.
29. w = __________
x = __________
y = __________
z = __________
perimeter = ____________________
30. p = __________
q = __________
r = __________
s = __________
perimeter = ____________________
Chapter 9: Area
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
Calculate the areas of the following shapes.
o A=bh
for a parallelogram or rectangle, where base and height form a right
angle
o A = s2
for a square
o A=½bh
for a triangle
o A = ½(b1 + b2)h for a trapezoid
o A = ½ d 1 d2
for a rhombus or a kite, where d = diagonal
o A=½Pa
for a regular polygon, where a = apothem, and P = perimeter
o Composite shapes
Find arc lengths and sector areas of circles.
o
o
Circumference = 2r
Area =  r2
Exercises for Area:
Use EXACT answers.
31. __________ Find the area of an equilateral triangle with side length of 6 ft.
32. __________ regular hexagon with side length of 4 cm.
33. __________ Find the area of an isosceles triangle with legs each 20 ft. long and a base 24 ft. long.
34. Area = __________
35. Length of arc PQ = __________
36. _______% Probability that a
Area of sector PNQ = __________ random point is in shaded region
Find the area of the figures below.
37. __________
38. __________
39. __________
40. __________
Chapter 10 Surface Area and Volume

Calculate the lateral area, total surface area, and volume of 3-D figures.
o Prisms and Cylinders
 LA = Ph
where P = perimeter of the base.
 SA = Ph + 2B
h = distance between the two bases.
 V = Bh
B = area of the base, which depends on the SHAPE of the base.
o Pyramids and Cones
 LA = 12 P ℓ
or
LA =  r ℓ for cones.
o
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
SA =

V=
1
3
1
2
Pℓ+B
where ℓ = slant height along a lateral face.
Bh
h = perpendicular height from the base to the vertex.
Sphere
 SA = 4  r2
 V = 43  r3
 A hemisphere is half a sphere.
o Composite figures: be careful when calculating surface area – do not include a base area if
it lies inside the figure. Break it down into parts.
o Oblique figures use the same formulas as right figures.
Use ratios of similarity, area, and volume to solve similar solids.
Exercises for Surface Area and Volume:
Find the surface area (SA) and volume (V) of each figure to the nearest tenth.
41. SA = __________
V = __________
42. SA = __________
V = ___________
43. SA = __________44. SA = __________
V = ___________
V = __________
45. __________ Two similar cones have heights of 9 cm and 4 cm. Find the ratio of their volumes.
46. a) __________ A cylinder a surface area of 25 cm2 , if the dimensions are double, what is the
surface area of the new cylinder?
b) __________ If the sphere has a volume of 50 cm3 and the dimensions are cut in half, what is
the volume of the larger sphere?
47. a)___________Find the entire figure’s surface area to the nearest tenth.
b) __________ Find the entire figure’s volume to the nearest tenth.
48. ______ Identify the cross section.
Chapter 12 Circles
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
Solve
o
o
o
o
Write
o
for missing lengths or angles using:
Properties of tangents
Properties of chords and arcs
Properties of inscribed angles and central angles
Relationships between secants and tangents
the standard equation for a circle in the coordinate plane
Circle equation is
(x – h)2 + (y – k)2 = r2
where (h, k) is the center and r is the radius
Exercises for Circles:
Find the measure of arc AB.
49. arc AB = ________ 50. arc AB = __________
51. a) _____ b) _____ c) _____ d) _______
Find the value of the variable(s). Assume that lines that appear to be tangent are tangent. Round to
the tenth.
52. __________
53. __________
56. x = ______ y= ______ 57. __________
54. __________
55. __________
58. __________
59. __________
60. __________ Write an equation of the circle that passes through (2, 8) with center (-3, 4).
Convert the following degree measure in radians.
61. __________ 90⁰
62.__________ 120⁰
63.__________ 330⁰
Chapter 13 Probability
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Geometric Probability:
o A ratio that involves geometric measurements
Probability
o Probability of an event occurring is the ratio of the number of favorable outcomes to the
number of possible outcomes
o Independent events are events where the outcome of one event does NOT affect the
outcome of the other events
Exercises for Probability:
64.______ What is the probability of the spinner landing on C?
65.______ What is the probability of NOT landing on C?
66.______ What is the probability of the spinner landing on A or B?
67._____ What is the probability of the spinner landing on one of the first five letters of the alphabet?
67.
68._____ What is the geometric probability of a dart landing in the shaded area?
69._____ What is the geometric probability of a dart landing in the unshaded area?