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New York Regents Geometry (Common Core) August 2015 Exam

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GEOMETRY (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY (Common Core) Wednesday, August 12, 2015 8:30 to 11:30 a.m., only Student Name: _________________________________________________________ School Name: _______________________________________________________________ The possession or use of any communications device is strictly prohibited when taking this examination. If you have or use any communications device, no matter how briefly, your examination will be invalidated and no score will be calculated for you. Print your name and the name of your school on the lines above. A separate answer sheet for Part I has been provided to you. Follow the instructions from the proctor for completing the student information on your answer sheet. This examination has four parts, with a total of 36 questions. You must answer all questions in this examination. Record your answers to the Part I multiple-choice questions on the separate answer sheet. Write your answers to the questions in Parts II, III, and IV directly in this booklet. All work should be written in pen, except for graphs and drawings, which should be done in pencil. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. The formulas that you may need to answer some questions in this examination are found at the end of the examination. This sheet is perforated so you may remove it from this booklet. Scrap paper is not permitted for any part of this examination, but you may use the blank spaces in this booklet as scrap paper. A perforated sheet of scrap graph paper is provided at the end of this booklet for any question for which graphing may be helpful but is not required. You may remove this sheet from this booklet. Any work done on this sheet of scrap graph paper will not be scored. When you have completed the examination, you must sign the statement printed at the end of the answer sheet, indicating that you had no unlawful knowledge of the questions or answers prior to the examination and that you have neither given nor received assistance in answering any of the questions during the examination. Your answer sheet cannot be accepted if you fail to sign this declaration. Notice A graphing calculator, a straightedge (ruler), and a compass must be available for you to use while taking this examination. DO NOT OPEN THIS EXAMINATION BOOKLET UNTIL THE SIGNAL IS GIVEN. GEOMETRY (COMMON CORE) Part I Answer all 24 questions in this part. Each correct answer will receive 2 credits. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For each statement or question, choose the word or expression that, of those given, best completes the statement or answers the question. [48] Use this space for computations. 1 A parallelogram must be a rectangle when its (1) diagonals are perpendicular (2) diagonals are congruent (3) opposite sides are parallel (4) opposite sides are congruent 2 If A B C is the image of ABC, under which transformation will the triangles not be congruent? (1) reflection over the x-axis (2) translation to the left 5 and down 4 (3) dilation centered at the origin with scale factor 2 (4) rotation of 270 counterclockwise about the origin 3 If the rectangle below is continuously rotated about side w, which solid figure is formed? w (1) pyramid (3) cone (2) rectangular prism (4) cylinder 4 Which expression is always equivalent to sin x when 0 x 90 ? (1) cos (90 x) (3) cos (2x) (2) cos (45 x) (4) cos x Geometry (Common Core) Aug. 15 [2] 5 In the diagram below, a square is graphed in the coordinate plane. Use this space for computations. y x A reflection over which line does not carry the square onto itself? (1) x 5 (3) y x (2) y 2 (4) x y 4 6 The image of ABC after a dilation of scale factor k centered at point A is ADE, as shown in the diagram below. A B C E D Which statement is always true? (1) 2AB AD (2) AD DE Geometry (Common Core) Aug. 15 (3) AC CE (4) BC || DE [3] [OVER] 7 A sequence of transformations maps rectangle ABCD onto rectangle A B C D , as shown in the diagram below. y A B D C x C D D A C A B B Which sequence of transformations maps ABCD onto A B C D and then maps A B C D onto A B C D ? (1) a reflection followed by a rotation (2) a reflection followed by a translation (3) a translation followed by a rotation (4) a translation followed by a reflection 8 In the diagram of parallelogram FRED shown below, ED is extended to A, and AF is drawn such that AF DF . F A R D E If m R 124 , what is m AFD? (1) 124 (3) 68 (2) 112 (4) 56 Geometry (Common Core) Aug. 15 [4] Use this space for computations. x2 y2 9 If 4x 6y 12 0 is the equation of a circle, the length of the radius is (1) 25 (3) 5 (2) 16 Use this space for computations. (4) 4 10 Given MN shown below, with M( 6,1) and N(3, 5), what is an equation of the line that passes through point P(6,1) and is parallel to MN ? y M P x N 2 3 (3) y 3 x 7 2 2 3 (4) y 3 x 8 2 (1) y x 5 (2) y x 3 11 Linda is designing a circular piece of stained glass with a diameter of 7 inches. She is going to sketch a square inside the circular region. To the nearest tenth of an inch, the largest possible length of a side of the square is (1) 3.5 (3) 5.0 (2) 4.9 (4) 6.9 Geometry (Common Core) Aug. 15 [5] [OVER] 12 In the diagram shown below, AC is tangent to circle O at A and to circle P at C, OP intersects AC at B, OA 4, AB 5, and PC 10. C 10 O B P 4 5 A What is the length of BC? (1) 6.4 (3) 12.5 (2) 8 (4) 16 13 In the diagram below, which single transformation was used to map triangle A onto triangle B? y B A x (1) line reflection (3) dilation (2) rotation (4) translation Geometry (Common Core) Aug. 15 [6] Use this space for computations. 14 In the diagram below, DEF is the image of ABC after a clockwise rotation of 180 and a dilation where AB 3, BC 5.5, AC 4.5, DE 6, FD 9, and EF 11. 11 F Use this space for computations. E 9 6 A 4.5 3 B D C 5.5 Which relationship must always be true? (1) m A 1 2 m D (3) m A m F (2) m C 2 1 m F (4) m B m C m C m D m E m F 15 In the diagram below, quadrilateral ABCD is inscribed in circle P. B A 72 110 P D C What is m ADC? (1) 70 (3) 108 (2) 72 (4) 110 Geometry (Common Core) Aug. 15 [7] [OVER] 16 A hemispherical tank is filled with water and has a diameter of 10 feet. If water weighs 62.4 pounds per cubic foot, what is the total weight of the water in a full tank, to the nearest pound? (1) 16,336 (3) 130,690 (2) 32,673 (4) 261,381 17 In the diagram below, ABC ADE. A D E B C Which measurements are justified by this similarity? (1) AD 3, AB 6, AE 4, and AC 12 (2) AD 5, AB 8, AE 7, and AC 10 (3) AD 3, AB 9, AE 5, and AC 10 (4) AD 2, AB 6, AE 5, and AC 15 18 Triangle FGH is inscribed in circle O, the length of radius OH is 6, and FH OG. F 6 H O G What is the area of the sector formed by angle FOH? (1) 2 (2) 3 2 Geometry (Common Core) Aug. 15 (3) 6 (4) 24 [8] Use this space for computations. ___ ___ 19 As shown in the diagram below, AB and CD intersect at E, and AC || BD. Use this space for computations. A C E D B Given AEC BED, which equation is true? (1) CE EB DE EA (3) EC BE AE ED (2) AE AC BE BD (4) ED AC EC BD 20 A triangle is dilated by a scale factor of 3 with the center of dilation at the origin. Which statement is true? (1) The area of the image is nine times the area of the original triangle. (2) The perimeter of the image is nine times the perimeter of the original triangle. (3) The slope of any side of the image is three times the slope of the corresponding side of the original triangle. (4) The measure of each angle in the image is three times the measure of the corresponding angle of the original triangle. 21 The Great Pyramid of Giza was constructed as a regular pyramid with a square base. It was built with an approximate volume of 2,592,276 cubic meters and a height of 146.5 meters. What was the length of one side of its base, to the nearest meter? (1) 73 (3) 133 (2) 77 (4) 230 Geometry (Common Core) Aug. 15 [9] [OVER] 22 A quadrilateral has vertices with coordinates ( 3,1), (0,3), (5,2), and ( 1, 2). Which type of quadrilateral is this? (1) rhombus (3) square (2) rectangle (4) trapezoid 23 In the diagram below, ABE is the image of ACD after a dilation centered at the origin. The coordinates of the vertices are A(0,0), B(3,0), C(4.5,0), D(0,6), and E(0,4). y D E A B x C The ratio of the lengths of BE to CD is (1) 2 (3) 3 (2) 3 (4) 4 3 2 4 3 24 Line y 3x 1 is transformed by a dilation with a scale factor of 2 and centered at (3,8). The line s image is (1) y 3x 8 (3) y 3x 2 (2) y 3x 4 (4) y 3x 1 Geometry (Common Core) Aug. 15 [10] Use this space for computations. Part II Answer all 7 questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil. [14] 25 A wooden cube has an edge length of 6 centimeters and a mass of 137.8 grams. Determine the density of the cube, to the nearest thousandth. State which type of wood the cube is made of, using the density table below. Type of Wood Density (g/cm3) Pine Hemlock 0.431 Elm 0.554 Birch 0.601 Ash 0.638 Maple 0.676 Oak Geometry (Common Core) Aug. 15 0.373 0.711 [11] [OVER] 26 Construct an equilateral triangle inscribed in circle T shown below. [Leave all construction marks.] T Geometry (Common Core) Aug. 15 [12] 27 To find the distance across a pond from point B to point C, a surveyor drew the diagram below. The measurements he made are indicated on his diagram. B 120 yd D A 230 yd E 85 yd C Use the surveyor s information to determine and state the distance from point B to point C, to the nearest yard. Geometry (Common Core) Aug. 15 [13] [OVER] 28 In parallelogram ABCD shown below, diagonals AC and BD intersect at E. D C E A B Prove: ACD CAB Geometry (Common Core) Aug. 15 [14] 29 Triangles RST and XYZ are drawn below. If RS 6, ST 14, XY 9, YZ 21, and S Y, is RST similar to XYZ? Justify your answer. X T Y S R Geometry (Common Core) Aug. 15 Z [15] [OVER] 30 In the diagram below, ABC and XYZ are graphed. y A B C x Y Z X Use the properties of rigid motions to explain why Geometry (Common Core) Aug. 15 [16] ABC XYZ. 31 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE:EF 2:3. Geometry (Common Core) Aug. 15 [17] [OVER] Part III Answer all 3 questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil. [12] 32 As shown in the diagram below, a ship is heading directly toward a lighthouse whose beacon is 125 feet above sea level. At the first sighting, point A, the angle of elevation from the ship to the light was 7 . A short time later, at point D, the angle of elevation was 16 . B 16 7 A D 125 ft C To the nearest foot, determine and state how far the ship traveled from point A to point D. Geometry (Common Core) Aug. 15 [18] 33 Triangle ABC has vertices with A(x,3), B( 3, 1), and C( 1, 4). Determine and state a value of x that would make triangle ABC a right triangle. Justify why ABC is a right triangle. [The use of the set of axes below is optional.] y x Geometry (Common Core) Aug. 15 [19] [OVER] 34 In the diagram below, AC DF and points A, C, D, and F are collinear on line . E A C D B F Let D E F be the image of DEF after a translation along , such that point D is mapped onto point A. Determine and state the location of F . Explain your answer. Let D E F be the image of D E F after a reflection across line . Suppose that E is located at B. Is DEF congruent to ABC? Explain your answer. Geometry (Common Core) Aug. 15 [20] Part IV Answer the 2 questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. All answers should be written in pen, except for graphs and drawings, which should be done in pencil. [12] 35 In the diagram of parallelogram ABCD below, BE CED, DF BFC, and CE CF . F B C E A D Prove ABCD is a rhombus. Geometry (Common Core) Aug. 15 [21] [OVER] 36 Walter wants to make 100 candles in the shape of a cone for his new candle business. The mold shown below will be used to make the candles. Each mold will have a height of 8 inches and a diameter of 3 inches. To the nearest cubic inch, what will be the total volume of 100 candles? Walter goes to a hobby store to buy the wax for his candles. The wax costs $0.10 per ounce. If the weight of the wax is 0.52 ounce per cubic inch, how much will it cost Walter to buy the wax for 100 candles? If Walter spent a total of $37.83 for the molds and charges $1.95 for each candle, what is Walter s profit after selling 100 candles? Geometry (Common Core) Aug. 15 [22] FOR TEACHERS ONLY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY (COMMON CORE) Wednesday, August 12, 2015 8:30 to 11:30 a.m., only SCORING KEY AND RATING GUIDE Mechanics of Rating The following procedures are to be followed for scoring student answer papers for the Regents Examination in Geometry (Common Core). More detailed information about scoring is provided in the publication Information Booklet for Scoring the Regents Examination in Geometry (Common Core). Do not attempt to correct the student s work by making insertions or changes of any kind. In scoring the open-ended questions, use check marks to indicate student errors. Unless otherwise specified, mathematically correct variations in the answers will be allowed. Units need not be given when the wording of the questions allows such omissions. Each student s answer paper is to be scored by a minimum of three mathematics teachers. No one teacher is to score more than approximately one-third of the open-ended questions on a student s paper. Teachers may not score their own students answer papers. On the student s separate answer sheet, for each question, record the number of credits earned and the teacher s assigned rater/scorer letter. Schools are not permitted to rescore any of the open-ended questions on this exam after each question has been rated once, regardless of the final exam score. Schools are required to ensure that the raw scores have been added correctly and that the resulting scale score has been determined accurately. Raters should record the student s scores for all questions and the total raw score on the student s separate answer sheet. Then the student s total raw score should be converted to a scale score by using the conversion chart that will be posted on the Department s web site at: http://www.p12.nysed.gov/assessment/ on Wednesday, August 12, 2015. Because scale scores corresponding to raw scores in the conversion chart may change from one administration to another, it is crucial that, for each administration, the conversion chart provided for that administration be used to determine the student s final score. The student s scale score should be entered in the box provided on the student s separate answer sheet. The scale score is the student s final examination score. If the student s responses for the multiple-choice questions are being hand scored prior to being scanned, the scorer must be careful not to make any marks on the answer sheet except to record the scores in the designated score boxes. Marks elsewhere on the answer sheet will interfere with the accuracy of the scanning. Part I Allow a total of 48 credits, 2 credits for each of the following. Allow credit if the student has written the correct answer instead of the numeral 1, 2, 3, or 4. (1) . . . . . 2 . . . . . (9) . . . . . 3 . . . . . (17) . . . . . 4 . . . . . (2) . . . . . 3 . . . . . (10) . . . . . 1 . . . . . (18) . . . . . 3 . . . . . (3) . . . . . 4 . . . . . (11) . . . . . 2 . . . . . (19) . . . . . 2 . . . . . (4) . . . . . 1 . . . . . (12) . . . . . 3 . . . . . (20) . . . . . 1 . . . . . (5) . . . . . 1 . . . . . (13) . . . . . 2 . . . . . (21) . . . . . 4 . . . . . (6) . . . . . 4 . . . . . (14) . . . . . 4 . . . . . (22) . . . . . 4 . . . . . (7) . . . . . 1 . . . . . (15) . . . . . 3 . . . . . (23) . . . . . 1 . . . . . (8) . . . . . 3 . . . . . (16) . . . . . 1 . . . . . (24) . . . . . 4 . . . . . Updated information regarding the rating of this examination may be posted on the New York State Education Department s web site during the rating period. Check this web site at: http://www.p12.nysed.gov/assessment/ and select the link Scoring Information for any recently posted information regarding this examination. This site should be checked before the rating process for this examination begins and several times throughout the Regents Examination period. Beginning in June 2015, the Department is providing supplemental scoring guidance, the Model Response Set, for the Regents Examination in Geometry (Common Core). This guidance is intended to be part of the scorer training. Schools should use the Model Response Set along with the rubrics in the Scoring Key and Rating Guide to help guide scoring of student work. While not reflective of all scenarios, the Model Response Set illustrates how less common student responses to constructed-response questions may be scored. The Model Response Set will be available on the Department s web site at: http://www.nysedregents.org/Geometrycc/. Geometry (Common Core) Rating Guide Aug. 15 [2] Question 25 25 A wooden cube has an edge length of 6 centimeters and a mass of 137.8 grams. Determine the density of the cube, to the nearest thousandth. State which type of wood the cube is made of, using the density table below. Type of Wood Density (g/cm3) Pine Hemlock 0.431 Elm 0.554 Birch 0.601 Ash 0.638 Maple 0.676 Oak Score 2: 0.373 0.711 The student has a complete and correct response. Geometry (Common Core) Aug. 15 [2] Question 26 26 Construct an equilateral triangle inscribed in circle T shown below. [Leave all construction marks.] Score 2: The student drew a correct construction showing all appropriate construction marks. Geometry (Common Core) Aug. 15 [5] Question 27 27 To find the distance across a pond from point B to point C, a surveyor drew the diagram below. The measurements he made are indicated on his diagram. B 120 yd D A 230 yd E 85 yd C Use the surveyor s information to determine and state the distance from point B to point C, to the nearest yard. Score 2: The student has a complete and correct response. Geometry (Common Core) Aug. 15 [9] Question 28 28 In parallelogram ABCD shown below, diagonals AC and BD intersect at E. D C E A B Prove: ACD CAB Score 2: The student has written a complete and correct proof. Geometry (Common Core) Aug. 15 [15] Question 29 29 Triangles RST and XYZ are drawn below. If RS 6, ST 14, XY 9, YZ 21, and S Y, is RST similar to XYZ? Justify your answer. X T Y S Score 2: R Z The student has a complete and correct response. Geometry (Common Core) Aug. 15 [18] Question 30 30 In the diagram below, ABC and XYZ are graphed. y A B C x Y Z X Use the properties of rigid motions to explain why Score 2: ABC The student has a complete and correct response. Geometry (Common Core) Aug. 15 [23] XYZ. Question 31 31 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE:EF 2:3. Score 2: The student has a complete and correct response. Geometry (Common Core) Aug. 15 [27] Question 32 32 As shown in the diagram below, a ship is heading directly toward a lighthouse whose beacon is 125 feet above sea level. At the first sighting, point A, the angle of elevation from the ship to the light was 7 . A short time later, at point D, the angle of elevation was 16 . B 16 7 A D 125 ft C To the nearest foot, determine and state how far the ship traveled from point A to point D. Score 4: The student has a complete and correct response. Geometry (Common Core) Aug. 15 [34] Question 33 33 Triangle ABC has vertices with A(x,3), B( 3, 1), and C( 1, 4). Determine and state a value of x that would make triangle ABC a right triangle. Justify why ABC is a right triangle. [The use of the set of axes below is optional.] Score 4: The student has a complete and correct response. Geometry (Common Core) Aug. 15 [42] Question 34 34 In the diagram below, AC DF and points A, C, D, and F are collinear on line . E A C D B F Let D E F be the image of DEF after a translation along , such that point D is mapped onto point A. Determine and state the location of F . Explain your answer. Let D E F be the image of D E F after a reflection across line . Suppose that E is located at B. Is DEF congruent to ABC? Explain your answer. Score 4: The student has a complete and correct response. Geometry (Common Core) Aug. 15 [50] Question 35 35 In the diagram of parallelogram ABCD below, BE CED, DF BFC, and CE CF . Prove ABCD is a rhombus. Score 6: The student has a complete and correct proof. Geometry (Common Core) Aug. 15 [58] Question 36 36 Walter wants to make 100 candles in the shape of a cone for his new candle business. The mold shown below will be used to make the candles. Each mold will have a height of 8 inches and a diameter of 3 inches. To the nearest cubic inch, what will be the total volume of 100 candles? Walter goes to a hobby store to buy the wax for his candles. The wax costs $0.10 per ounce. If the weight of the wax is 0.52 ounce per cubic inch, how much will it cost Walter to buy the wax for 100 candles? If Walter spent a total of $37.83 for the molds and charges $1.95 for each candle, what is Walter s profit after selling 100 candles? Score 6: The student has a complete and correct response. Geometry (Common Core) Aug. 15 [68]

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