2019-2020 Geometry Summer Assignment

You must show all work to earn full credit. This assignment will be due Friday, August 23, 2019. 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

3x 33. 513

3

2x

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