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Midterm Exam Review 34 questions Slides 2 - 14 Geometry Honors Mrs. Tiesi Warm-up (12/7 – 12/11) Chapter 3 1) Write an equation in slope-intercept form for a line perpendicular to a) y = (3/4)x + 2 containing (6, 5). b) y = (1/3)x + 2 containing (8, 9). 2) Determine whether lines AB and CD are parallel, perpendicular, or neither. a) A(1, 5), B(4, 4), C(9, -10), D(-6, -5) b) A(4, 2), B(-3, 1), C(6, 0), D(-10, 8) 3) Find the distance between each pair of parallel lines with the given equations. a) y = 5x - 22 and y = 5x + 4 b) y = (1/3)x - 3 and y = (1/3)x + 2 Hint: Find the slope-intercept equation of line p (perpendicular to line a at the same y-intercept (0, b)). Solve a sytem of equations (to find an intercepted point (x, y)) using the equations of lines b and p. Use the Distance Formula for 2 points (0, b) and (x, y) to find the answer. 4) Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. a) <5 and <16; <6 and <13; <9 and <13; <4 and <5 b) <7 and <2; <11 and <15; <3 and <6; <4 and 11 m n d c Chapters 1 & 4 2-column & Flow proof 5) Sec 2-6: Write a 2-column proof a) If -4(x - 3) + 5x = 24, then x = 12. b) If (8 – 3x) / 4 = 32, then x = -40. 6) (Sec 4-4 and 4-5) Write a flow proof a) Given: BA = DC, <BAC = <DCA Prove: BC = DA b) Given: R is the midpoint of QS, <PQR = <TSR Prove: ∆PRS = ∆TRS A B D C S P R Q T B C 7) Given: AB = DC; AD = BC = D Prove: ∆ABD ˜ ∆CDB A X Y B Z 8) Given: ∆ZBX ˜= ∆WBY <ZXY = <WYX Prove: ∆ZXY ˜= ∆WYX W Points, lines and planes 9) Name a point on line m. B 10) Using AI and FE, Name a pair of acute vertical angles Name a pair of obtuse vertical angles F H D C A G E m H V 11) Identify a parallel plane to plane QWM. Identify all intersecting planes to plane AVS. 12) Identify all skew lines to AV. I S A W Q M P Volumes/Surface Areas of all solids 13) Baseballs and softballs come in different sizes for different types of leagues. If the diameter of a baseball is 5 inches and a softball has a diameter of 5.4 inches, find the difference between the volumes of the two balls. Round to the nearest tenth (V = 4πr3/3). 14) Cakes are stacked in 2 layers as a cylinder. If the radius of a cake is 10 inches and its height is 12 inches, find the surface area of the frosting (T = 2πrh + 2πr2). 15) A rectangular prism pool needs to be painted. If the bottom of the pool has dimensions 15 ft x 20 ft, and its height is 8ft, find the paint area. (Paint Area = Ph + B ; P = 2l + 2w ; B = l ● w) Chapter 1 Distance, Midpoint and Slope Formulas 16) a) A segment has a midpoint at (3, 9) and an endpoint at (14, -8), what is the other endpoint? b) Find the distance between 2 points: (3, 9) and (14, -8). c) Find the slope between 2 points: (3, 9) and (14, -8) Triangle Problems 17) Use Angle Sum Theorem to find <A of ∆ABC, given <B = 750 and <C = 350. 18) Classify the triangle by its angles. <A = 10º, <B = 160º, <C = 10º <A = 60º, <B = 60º, <C = 60º <A = 70º, <B = 50º, <C = 60º <A = 60º, <B = 30º, <C = 90º 19) Classify the triangle by its sides. AB = 1/3 cm , BC = 1/3 cm, AC = ½ cm AB = 8 in, BC = 10 in, AC = 12 in AB = 2.1 ft, BC = 2.1 ft, AC = 2.1ft 20) ∆ABC is an isosceles right triangle. <B is the vertex right angle. If AB = 4x – 1, BC = 2x + 5 and AC = 8x – 8.44, find x and all sides. Also, what is the measurement of <A or <C? 21) ∆ABC is an equilateral triangle. If <A = 4x, solve for x. If AB = 3y – 10, BC = 2y + 5, and AC = y + 20, find y and all sides. Chapter 1 Polygons 22) Identify the polygon names, concave/convex, regular/irregular Chapter 4 Exterior Angle Theorem 23) Use Exterior Angle 1 Theorem, If <3 = 80º, <6 = 108º, find <2. If <7 = 30º, <5 = 70º, find <4. 3 5 2 4 6 7 8 The rest of the problems 24) Given <1 and <2: State the theorems/postulates (Complement, Supplement, Vertical, Corresponding, Alternate Interior, Alternate Exterior, Consecutive Interior angles) for each of the following statement: <1 = <2 <1 + <2 = 90º <1 + <2 = 180º 25) Find the Volume and Surface Area of a cylinder, given r = 10cm, h = 8cm. 26) Find the Volume and Surface Area of a rectangular prism, given length = 11cm, width = 7cm and height = 5cm. 27) Find the base of a triangle, given h = 10cm and A = 200cm2. 28) Find the width of a rectangle, given w = 11 in and A = 132 in2. 29) Find the diagonal of a square, given A = 100cm2 30) Write a converse, inverse and contrapositive of the following statement: If you are a dancer, then you love to dance. 31) Identify the number properties: a) A number is equal to itself. b) If 6 = x, then x = 6. c) If x = 3 + 5 and 3 + 5 = 8, then x = 8. d) If x = 3 + 5 and 8 = 3 + 5, then x = 8. Chapter 4 32) a) Position and label an isosceles triangle ABC with a base BC of 2a. b) Position and label an equilateral triangle ABC with a side of 2a. 33) Prove 3 points form a right triangle P(0, 0), N(3b, 0), M(0, 2a) • Find the slopes of all 3 sides. • Check if 2 sides form perpendicular lines. – 1 slope is zero and the other is undefined. – 1 slope is the reverse reciprocal of the other. Chapter 9 34) Given an original point of (2, -4) a) Reflection over the x-axis, y-axis, y = x, y = 1, x = -2. b) Translation using a component vector <3, 2> c) Rotation 90, 180 and 270 counterclockwise about the origin. Notes Chapter 4 Proving triangles congruent a) 5 methods: Congruent Triangles: all corresponding angles and sides are congruent. SSS: 3 pairs of corresponding sides SAS: 2 pairs of corresponding sides, the included angle is between 2 sides. ASA: 2 pairs of corresponding angles, the included side is between the 2 angles. AAS: 2 angles and 1 non-included side. b) CPCTC: Corresponding parts of congruent triangles are congruent. A C B 1 2 E SAS Ex: Given: AB = BD EB = BC Prove: ∆ABE = ˜ ∆DBC Our Outline P rerequisites D S ides A ngles S ides Triangles =˜ C Given: CX bisects ACB A ˜ B = 12 Prove: ∆ACX ˜=∆BCX AAS A X B P CX bisects ACB A 1= 2 A A= B S CX = CX ∆’s ∆ACX =˜ ∆BCX Given Def of angle bisc Given Reflexive Prop AAS POLYGON NAMES NAMES TRIANGLE QUADRILATERAL PENTAGON HEXAGON HEPTAGON OCTAGON NONAGON DECAGON HENDECAGON DODECAGON N – GON #SIDES 3 4 5 6 7 8 9 10 11 12 N SEE PAGE 57 IN TEXTBOOK CONCAVE CONVEX Regular or Irregular polygons • An Equilateral polygon: all sides are congruent. • An Equiangular polygon: all angles are congruent. • A regular polygon: a convex polygon with all congruent sides and angles. • An irregular polygon: is a polygon that is not regular. Isosceles, Equilateral Triangles vertex angle A B leg leg base angles A base C B C Area of a Triangle A = ½ bh. A = ½ (23 ft)(6 ft) A = 69 ft 2 6 ft. Height Base 23 ft. A = ½ bh. Area of a Square and a Rectangle width = 10cm side = 8cm length = 18cm A = s2 A=l●w A = (8cm)(8cm) A= 64cm2 A = (18cm)(10cm) A = 180cm2 Find L, B, S and V L = 2rh = 2(3.14)(6)(9) 6ft = 339.12ft2 B = r2 = 3.14(6)2 = 113.04ft2 9ft S = L + 2B = 339.12ft2 + 2 ●113.04ft2 = 565.2ft2 V = Bh = 113.04 ● 9 = 1017.4ft3 Find P, B, L, S and V of a prism Base shape is a rectangle P = 2l + 2w L = P●h 5cm B = l●w S = L + 2B 3cm 4cm V = B●h Important Properties • • • • Reflexive : a = a Symmetric : if a = b then b = a Transitive : if a = b and b = c then a = c Substitution : if a = b then a may be used in any equation instead of b • Distributive: a(b + c) = ab + ac • Addition and Subtraction if a = b then a+c = b+c and a-c = b-c • Multiplication and Division if a = b then ac = bc and a / c = b / c