2018-2019 geometry summer assignment all work...2018-2019 geometry summer assignment you must show...
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2018-2019 Geometry Summer Assignment
You must show all work to earn full credit. This assignment will be due Friday, August 24, 2018. It will be worth
50 points. All of these skills are necessary to be successful in Geometry Honors. Geometry is a branch of
mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
However, in order to solve those questions, algebra will be crucial. There will be a test on these topics at the
beginning of the year. Please follow criteria for credit shown below. You may write on the back of your
notebook paper. Do NOT write on this worksheet. -Mrs. Fussner
Section 1: Expressions & Equations A. Integer Operations - Integers are positive & negative whole numbers. No Calculators
Addition Subtraction Multiplication Division
Same Signs + and + – and –
• Add the numbers
• Take the sign of both numbers
• Subtract the numbers
• Take the sign of the number with the larger absolute value
• Multiply the numbers
• The product is always positive.
• Divide the numbers
• The quotient is always positive.
Examples 6 + 8 = 14 (–4) + (–5) = –9
10 – 15 = –5 (–7) – (–15) = (–7) + 15 = 8
(7)(8) = 56 (–11)( –12) = 132
9/3 = 3 (–72)/( –8) = 9
Different signs
+ and – – and +
• Subtract the numbers
• Take the sign of the number with the larger absolute value
• Add the numbers
• Take the sign of the number with the larger absolute value
• Multiply the numbers
• The product is always negative
• Divide the numbers
• The quotient is always negative.
Examples 14 + (–24) = –10 (–9) + 10 = 1
10 – (–5) = 10 + 5 = 15 (–8) – 7 = –15
(8)( –12) = –96 (–12)(10) = –120
9/(–3) = –3 (–72)/( 8) = –9
1. –2 + 3
2. –5 + 4
3. –7 – (–3)
4. –14 – 6
5. 6 + (–8)
6. 12 + (–7)
7. –8 + (–1)
8. –3(–4)
9. 24
−6
10. 5(–18)
11. 17(–4)
12. −21
−7
13. 81
−9
14. 45 – (–27)
15. −8
−4
B. Order of Operations No Calculators
16. 18 – (–12 – 3) 19. –19 + (7 + 4)3
17. 20 – 4(32 – 6) 20. –3 + 2(–6 ÷ 3)2
18. –6(12 – 15) + 23 21. 4(–6) + 8 – (–2)
15 – 7 + 2
Evaluate each expression if a = 12, b = 9, and c = 4.
22. 4a + 2b – c2 23. 2c3− ab
4 24. 2(a – b)2 – 5c
C. Solving Equations - Solve each equation for x. You may use a calculator but MUST show every step.
25. –20 = –4x – 6x 27. 8x – 2 = –9 + 7x 29. 4x + (5x – 36) = 90
26. 12 = –4(–6x – 3) 28. –3(4x + 3) + 4(6x + 1) = 43 30. (3x – 5) + (2x – 10) = 180
Example:
D. Solving Equations by Clearing the Fraction You may use a calculator but MUST show every step.
31. xxx2
13
4
1+−=+ 32. ( ) 212
4
3=+x 33. ( ) 513
3
2=+x
E. Solving Systems of Equations – Systems may have zero solutions, one solution or infinitely many solutions.
Solve the following systems of equations by substitution. You may use a calculator but MUST show every step.
34. x + 12y = 68 35. 3x + 2y = 6
x = 8y – 12 x – 2y = 10
Solve the following systems of equations by elimination. You may use a calculator but MUST show every step.
36. 2x + 5y = -4 37. 10x + 6y = 0
3x – y = 11 -7x + 2y = 31
Elimination Example:
Section 2: The Coordinate Plane & Linear Functions
F. The Coordinate Plane
G. Slope
Find the slope in each problem.
51. (–7, 8) 52. (6, –9) (3, –9) 53. 2x + 3y = 6 54. x = 4
Tell what point is located at each ordered pair.
38. (3, –2)
39. (–7, –8)
40. (2, 3)
41. (–4, 4)
42. (–5, 5)
43. (–5, 0)
Write the ordered pair for each given point & name the
quadrant it is in.
44. E
45. M
46. C
47. 48. 49. 50.
H. Graphing & Writing Equations of Lines
Slope-intercept form → y = mx + b
Point-slope form → y – y1 = m(x – x1)
For each set of ordered pairs, calculate the slope and write the equation of the line passing through each of the
points in both slope-intercept & point-slope form. Then graph the equation.
55. (0, –3) and (5, –1) 56. (-1, 3) and (-4, 7) 57. (9, -2) and (9, 4)
I. Parallel & Perpendicular Lines
Write an equation that is parallel to the given equation.
58. y = 3x + 5 59. -2x + y = 3
Write an equation that is perpendicular to the given equation.
60. 52
1+= xy 61. 2y – 4x = 8
62. Choose the two parallel lines 63. Choose the two perpendicular lines
y - 6 = 3x y – 3x = -4 y + 10 = -4x y = 4x - 10
3y = -x y + 9 = -3x -4y = x 4y = x – 5
64. Write an equation in slope-intercept form for the line that passes through the point (8, -12) and is parallel to
94
3−−= xy
65. Write an equation in slope-intercept form for the line that passes through the point (4, -3) and is
perpendicular to y = 4x +5
J. Midpoint & Distance Formula
Use the midpoint and distance formulas to find the midpoint and distance between each pair of points:
66. (7, 11) and (-1, 5) 69. (2, 0) and (8, 6)
67. (-2, -1) and (3, 11) 70. (-2, -6) and (6, 9)
68. (-10, 2) and (-7, 6) 71. (-3, 2) and (6, 5)
Section 3: Radicals
Radicals or roots are the “opposite” operation of applying exponents. You will undo exponents by using a
radical.
√𝑥 − 43
Read as the “cube root of x – 4”
K. Perfect Squares
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, ….
*This means if you see any of these numbers under the radical you can quickly simplify it by finding the number
that multiplies by itself to get the number.
Simplify:
72. √169 74. √25 76. √36
73. √49 75. √225 77. √9
L. Non-Perfect Squares
When the number under the radical cannot be equally split – in this case we have to reduce it to its lowest terms.
Math Look Fors
√75
√25√3
Identify the largest perfect square that divides evenly into
the radicand
5√3 Take the square root of the perfect square radical and
leave the non perfect square under its radical
Simplify:
78. √200 80. √20 82. √8
79. √72 81. √125 83. √48
Index
Radicand
Radical
Symbol
M. Radicals with Variables
Math Look Fors
√𝑚5
√𝑚4√𝑚1
Break the variable down into it’s perfect square exponent
(even exponent) and remaining exponent (remember that
multiplying 2 bases that are the same means we add the
exponents)
𝑚2√𝑚 Take the square root of the perfect square variable by
dividing its exponent by 2 and keep the non perfect
square variable under its radical.
Simplify:
84. √36𝑥2 85. √20𝑡5 86. √𝑝7 87. √𝑥6𝑦5𝑧7
N. Rationalizing the Denominator
A radical cannot be in the denominator of a fraction, so in order to fix it we have to multiply both the numerator
and denominator by the radical in the denominator (or one that will create a perfect square). Example: 2
√3
Math Look Fors
2
√3 •
√3
√3
Multiply both the numerator and denominator by the
radical in the denominator
2√3
√9=
2√3
3
Simplify the radicals to solve
Simplify:
88. 5
√6 89.
9
√8 90.
√49
√2 91.
3
√18
Section 4: Proportions
O. Solving Proportions
Solve for x:
92. 𝑥
7=
10
14
93. 15
𝑥=
3
4
94. 𝑥
7=
10
14
95. 𝑥+1
10=
2
4
96. 5
6=
𝑥−2
𝑥+3
97. 2
3=
15
𝑥
P. Conversions
Convert to the given unit:
98. 5 ft. = _______in
99. 20 m = ________ cm
100. 450 cm = ________m
101. 48 in = ________ ft.
102. 14 ft2 = _________ in2
103. 288 in2 = _______ ft2
104. 50 m2 = ________cm2
105. 3000 cm2 = ________m2
Section 5: Polynomials
Q. Multiply Polynomials
*Distribute or use FOIL method.
(x + 5)(x + 7) = (x)(x) + (x)(7) + (5)(x) + (5)(7)
= x2 + 7x + 5x + 35
= x2 + 12x + 35
Find each product.
106. (r + 1)(r – 2) 108. (n – 5)(n + 1)
107. (3c + 1)(c – 2) 109. (2x – 6)(x + 3)
R. Factoring
Factoring is the process of “un-doing” a polynomial. Factors are what items multiplied together to get a product.
First always check to see if you can factor out a GCF.
t2 + 8t + 12
Math Look Fors
t2 + 8t + 12
1•12, 2•6, 3•4 are factors of “c”
Identify the factors of “c”
6 and 2 can be added to get 8 or “b” Find the factors of “c” that add or subtract to equal “b”
(t )(t ) Create your factors
(t + )(t + ) Identify the signs that fit into factors
• If + and +, then factors are both +
• If + and –, then larger factor get + and smaller
factor gets –
• If – and – , then larger factor gets – and smaller
factors get +
• If – and +, then both factors are -
(t + 2)(t + 6) Plug in the numbers
Factor each polynomial if possible. If the polynomial cannot be factored using integers, write prime.
110. p2 + 9p + 20 113. g2 – 7g + 2
111. n2 + 3n – 18 114. y2 – 5y – 6
112. t2 + 9t – 5 115. 4r2 + 16r – 48
Product of
First Terms
Product of
Outer Terms Product of
Inner Terms Product of
Last Terms
Section 6: Solving Quadratic Equations
S. Solving Quadratic Equations by Factoring
Example: x2 – 6x + 8 = 0
Math Look Fors
x2 – 6x + 8 = 0
1•8, 2•4 are factors of “c”
Identify the factors of “c”
2 and 4 can be added to get 6 or “b” Find the factors of “c” that add or subtract to equal “b”
(x )(x ) Create your factors
(x – )(x – ) Identify the signs that fit into factors
(x – 2)(x – 4) = 0 Plug in the numbers
x – 2 = 0 x – 4 = 0 Set each factor equal to zero
x – 2 = 0 x – 4 = 0
+ 2 +2 + 4 +4
x = 2 x = 4
Solve for x in each problem using inverse operations
x can either have a value of 2 or 4 Identify the solution
Solve each equation by factoring. Check the solutions.
116. d2 + 7d + 10 = 0 117. y2 – 2y – 24 = 0
T. Solving Quadratic Equations by Completing the Square
Solve by completing the square:
118. x2 + 4x - 10 = 0
119. x2 + 10x - 4 = 0
120. x2 + 6x + 1 = 0 121. x2 - 12x + 30 = 0
U. Solving Quadratic Equations by Quadratic Formula
Solve using quadratic formula:
122. x2 - 11x + 7 = 0
123. x2 + 7x - 4 = 0
124. 3x2 - 12x - 9 = 0
125. x2 + 6x + 20 = 0
Section 7: Geometry Basics
V. Points, Lines & Planes
W. Angles
Name each angle 4 ways, classify it and give its exact measure using a protractor.
131. 132. 133.
126. Name any two line segments.
127. Which point is not coplanar (points all on the same
plane) with the points U and V?
128. Write a set of points with are collinear (points on the
same line).
129. Name a pair of opposite rays (have the same endpoint &
extend in opposite directions).
130. Name the plane in two ways.
𝑋𝑌̅̅ ̅̅ 𝑜𝑟 𝑌𝑋̅̅ ̅̅
𝐴𝐵 ⃡ 𝑜𝑟 𝐵𝐴 ⃡
𝑃𝑄
𝑃 𝑜𝑟 𝑝𝑜𝑖𝑛𝑡 𝑃
𝑃𝑙𝑎𝑛𝑒 𝐸𝐹𝐺 𝑜𝑟
Plane T
134. Name two acute vertical angles.
135. Name two obtuse vertical angles.
136. Name a pair of adjacent angles
137. Name a linear pair.
138. Name a pair of complementary angles.
139. Name an angle supplementary to FGE
X. The Pythagorean Theorem
Find the missing side length of the right triangle using The Pythagorean Theorem.
140. 142.
141. 143.
Y. Transformations
Name the type of transformation depicted in the diagram below. If a reflection, state the line of symmetry. Dashed figure (preimage)→solid figure (image)
144. 145. 146. 147.
8
6
c
a
24
26
10
6 x
6
3
10
x
Z. Perimeter, Area, Surface Area & Volume
Prism Pyramid Cylinder Cone Sphere
Lateral Area (LA) area of the sides
𝑷𝒉 𝟏𝟐⁄ 𝑷𝒍 𝟐𝝅𝒓𝒉 𝝅𝒓𝒍
Surface Area area of sides & bases
𝑳𝑨 + 𝟐𝑩 𝑳𝑨 + 𝑩 𝑳𝑨 + 𝟐𝑩 𝑳𝑨 + 𝑩 𝟒𝝅𝒓𝟐
Volume 𝑩𝒉 𝟏𝟑⁄ 𝑩𝒉 𝑩𝒉 𝟏
𝟑⁄ 𝑩𝒉 𝟒𝟑⁄ 𝝅𝒓𝟑
Determine which choice BEST describes the figure.
148. 149. 150. 151.
Find the surface area and volume of the figure.
152. 153. 154. 155.
Key:
𝑃 = 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑜𝑙𝑖𝑑
ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑜𝑙𝑖𝑑 (or altitude)
𝑙 = 𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 𝑜𝑟 𝑐𝑜𝑛𝑒 (𝑚𝑖𝑔ℎ𝑡 ℎ𝑎𝑣𝑒 𝑡𝑜 𝑢𝑠𝑒 𝑝𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑒𝑎𝑛 𝑡ℎ𝑒𝑜𝑟𝑒𝑚 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑡ℎ𝑖𝑠)
𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒
𝐵 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑜𝑙𝑖𝑑