# The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

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2 Part I Answer all 28 questions in this part. Each crect answer will receive 2 credits. No partial credit will be allowed. F each question, write on the separate answer sheet the numeral preceding the wd expression that best completes the statement answers the question. [56] 1 In the diagram below, _ AB, _ BC, and _ AC are tangents to circle O at points F, E, and D, respectively, AF = 6, CD = 5, and BE = 4. Use this space f computations. B F E O A D C What is the perimeter of ABC? (1) 15 (3) 30 (2) 25 (4) 60 _ 2 Quadrilateral MNOP is a trapezoid with MN _ OP. If M N O P is the image of MNOP after a reflection over the x-axis, which two sides of quadrilateral _ M N O P _ are parallel? (1) M N _ and O P _ (3) P M _ and O P _ (2) M N and N O (4) P M and N O Geometry January 11 [2]

3 3 In the diagram below of ABC, D is the midpoint of _ AB, and E is the midpoint of _ BC. Use this space f computations. B D E A 4x + 10 C If AC = 4x + 10, which expression represents DE? (1) x (3) 2x + 10 (2) 2x + 5 (4) 8x Which statement is true about every parallelogram? (1) All four sides are congruent. (2) The interi angles are all congruent. (3) Two pairs of opposite sides are congruent. (4) The diagonals are perpendicular to each other. Geometry January 11 [3] [OVER]

4 5 The diagram below shows a rectangular prism. Use this space f computations. B C A D F G E H Which pair of edges are segments of lines that are coplanar? (1) _ AB and _ DH (3) _ BC and _ EH (2) _ AE and _ DC (4) _ CG and _ EF 6 A line segment has endpoints A(7, 1) and B( 3,3). What are the codinates of the midpoint of _ AB? (1) (1,2) (3) ( 5,2) (2) (2,1) (4) (5, 2) Geometry January 11 [4]

5 7 What is the image of the point ( 5,2) under the translation T 3, 4? (1) ( 9,5) (3) ( 2, 2) (2) ( 8,6) (4) ( 15, 8) Use this space f computations. 8 When writing a geometric proof, which angle relationship could be used alone to justify that two angles are congruent? (1) supplementary angles (2) linear pair of angles (3) adjacent angles (4) vertical angles 9 Plane R is perpendicular to line k and plane D is perpendicular to line k. Which statement is crect? (1) Plane R is perpendicular to plane D. (2) Plane R is parallel to plane D. (3) Plane R intersects plane D. (4) Plane R bisects plane D. Geometry January 11 [5] [OVER]

6 10 The vertices of the triangle in the diagram below are A(7,9), B(3,3), and C(11,3). Use this space f computations. y A(7,9) B(3,3) C(11,3) x What are the codinates of the centroid of ABC? (1) (5,6) (3) (7,5) (2) (7,3) (4) (9,6) 11 Which set of numbers does not represent the sides of a right triangle? (1) {6, 8, 10} (3) {8, 24, 25} (2) {8, 15, 17} (4) {15, 36, 39} Geometry January 11 [6]

7 12 In the diagram below of rhombus ABCD, m C = 100. Use this space f computations. A B D 100 C What is m DBC? (1) 40 (3) 50 (2) 45 (4) In the diagram below of circle O, radius _ OC is 5 cm. Chd _ AB is 8 cm and is perpendicular to _ OC at point P. A C P B O What is the length of _ OP, in centimeters? (1) 8 (3) 3 (2) 2 (4) 4 Geometry January 11 [7] [OVER]

8 14 What is an equation of the line that passes through the point ( 2,3) and is parallel to the line whose equation is y = 3 2 x 4? (1) y = 2 3 x (3) y = 3 2 x (2) y = 2 3 x (4) y = 3 2 x + 6 Use this space f computations. 15 In scalene triangle ABC, m B = 45 and m C = 55. What is the der of the sides in length, from longest to shtest? (1) _ AB, _ BC, _ AC (3) _ AC, _ BC, _ AB (2) _ BC, _ AC, _ AB (4) _ BC, _ AB, _ AC 16 What is an equation of a circle with center (7, 3) and radius 4? (1) (x 7) 2 + (y + 3) 2 = 4 (2) (x + 7) 2 + (y 3) 2 = 4 (3) (x 7) 2 + (y + 3) 2 = 16 (4) (x + 7) 2 + (y 3) 2 = 16 Geometry January 11 [8]

9 17 What is the volume, in cubic centimeters, of a cylinder that has a height of 15 cm and a diameter of 12 cm? (1) 180π (3) 675π (2) 540π (4) 2,160π Use this space f computations. 18 Which compound statement is true? (1) A triangle has three sides and a quadrilateral has five sides. (2) A triangle has three sides if and only if a quadrilateral has five sides. (3) If a triangle has three sides, then a quadrilateral has five sides. (4) A triangle has three sides a quadrilateral has five sides. 19 The two lines represented by the equations below are graphed on a codinate plane. x + 6y = 12 3(x 2) = y 4 Which statement best describes the two lines? (1) The lines are parallel. (2) The lines are the same line. (3) The lines are perpendicular. (4) The lines intersect at an angle other than 90. Geometry January 11 [9] [OVER]

10 20 Which diagram shows the construction of the perpendicular bisect of _ AB? Use this space f computations. A B A B (1) (3) A B A B (2) (4) 21 In circle O, a diameter has endpoints ( 5,4) and (3, 6). What is the length of the diameter? (1) 2 (3) 10 (2) 2 2 (4) 2 41 Geometry January 11 [10]

11 22 In the diagram below of quadrilateral ABCD, _ AB _ CD, ABC CDA, and diagonal _ AC is drawn. Use this space f computations. A B D C Which method can be used to prove that ABC is congruent to CDA? (1) AAS (3) SAS (2) SSA (4) SSS 23 In the diagram below of right triangle ABC, _ CD is the altitude to hypotenuse _ AB, CB = 6, and AD = 5. C 6 A 5 D B What is the length of _ BD? (1) 5 (3) 3 (2) 9 (4) 4 Geometry January 11 [11] [OVER]

12 24 In the diagram below, quadrilateral JUMP is inscribed in a circle. Use this space f computations. J U P M Opposite angles J and M must be (1) right (3) congruent (2) complementary (4) supplementary Geometry January 11 [12]

13 25 Which graph represents a circle with the equation (x 3) 2 + (y + 1) 2 = 4? Use this space f computations. y y x x (1) (3) y y x x (2) (4) 26 The point (3, 2) is rotated 90 about the igin and then dilated by a scale fact of 4. What are the codinates of the resulting image? (1) ( 12,8) (3) (8,12) (2) (12, 8) (4) ( 8, 12) Geometry January 11 [13] [OVER]

14 27 In the diagram below of ABC, side _ BC is extended to point D, m A = x, m B = 2x + 15, and m ACD = 5x + 5. Use this space f computations. A D C B What is m B? (1) 5 (3) 25 (2) 20 (4) Point P lies on line m. Point P is also included in distinct planes Q, R, S, and T. At most, how many of these planes could be perpendicular to line m? (1) 1 (3) 3 (2) 2 (4) 4 Geometry January 11 [14]

15 Part II Answer all 6 questions in this part. Each crect answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate fmula substitutions, diagrams, graphs, charts, etc. F all questions in this part, a crect numerical answer with no wk shown will receive only 1 credit. All answers should be written in pen, except f graphs and drawings, which should be done in pencil. [12] _ 29 In the diagram below of ACD, B is a point _ on _ AC such that ADB is an equilateral triangle, and DBC is an isosceles triangle with DB BC. Find m C. D A B C Geometry January 11 [15] [OVER]

16 30 Triangle ABC has vertices A( 2,2), B( 1, 3), and C(4,0). Find the codinates of the vertices of A B C, the image of ABC after the transfmation r x-axis. [The use of the grid below is optional.] Geometry January 11 [16]

17 31 Find, in degrees, the measures of both an interi angle and an exteri angle of a regular pentagon. Geometry January 11 [17] [OVER]

18 32 In the diagram below of circle O, chd _ AB bisects chd _ CD at E. If AE = 8 and BE = 9, find the length of _ CE in simplest radical fm. A C 9 E O 8 D B Geometry January 11 [18]

19 33 On the diagram below, use a compass and straightedge to construct the bisect of ABC. [Leave all construction marks.] A B C Geometry January 11 [19] [OVER]

20 34 Find the slope of a line perpendicular to the line whose equation is 2y 6x = 4. Geometry January 11 [20]

21 Part III Answer all 3 questions in this part. Each crect answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate fmula substitutions, diagrams, graphs, charts, etc. F all questions in this part, a crect numerical answer with no wk shown will receive only 1 credit. All answers should be written in pen, except f graphs and drawings, which should be done in pencil. [12] 35 On the set of axes below, graph the locus of points that are four units from the point (2,1). On the same set of axes, graph the locus of points that are two units from the line x = 4. State the codinates of all points that satisfy both conditions. y x Geometry January 11 [21] [OVER]

22 36 In the diagram below, _ BFCE, _ AB _ BE, _ DE _ BE, and BFD ECA. Prove that ABC DEF. A D B F C E Geometry January 11 [22]

23 37 In the diagram below of ADE, B is a point on _ AE and C is a point on _ AD such that _ BC _ ED, AC = x 3, BE = 20, AB = 16, and AD = 2x + 2. Find the length of _ AC. A B C E D Geometry January 11 [23] [OVER]

24 Part IV Answer the question in this part. A crect answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate fmula substitutions, diagrams, graphs, charts, etc. A crect numerical answer with no wk shown will receive only 1 credit. The answer should be written in pen. [6] 38 Quadrilateral MATH has codinates M(1,1), A( 2,5), T (3,5), and H(6,1). Prove that quadrilateral MATH is a rhombus and prove that it is not a square. [The use of the grid on the next page is optional.] Geometry January 11 [24]

25 Question 38 continued Geometry January 11 [25]

26

27 Reference Sheet Tear Here Tear Here Cylinder V Bh where B is the area of the base Volume Pyramid Right Circular Cone V 1 Bh 3 where B is the area of the base V 1 Bh 3 where B is the area of the base Sphere V 4 r 3 3 Lateral Area (L) Right Circular Cylinder Right Circular Cone L 2 rh L rl where l is the slant height Surface Area Sphere SA 4 r 2 Geometry January 11 [27]

28 Tear Here Tear Here

29 Scrap Graph Paper This sheet will not be sced. Tear Here Tear Here

30 Scrap Graph Paper This sheet will not be sced. Tear Here Tear Here

32 GEOMETRY Rater s/scer s Name (minimum of three) GEOMETRY Question Maximum Credit Part I Part II Part III Part IV 38 6 Maximum 86 Total Credits Earned Total Raw Sce Rater s/scer s Initials Checked by Scale Sce (from conversion chart) Tear Here Tear Here Geometry January 11 [32] Printed on Recycled Paper GEOMETRY

33 FOR TEACHERS ONLY The University of the State of New Yk REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 27, :15 a.m. to 12:15 p.m., only SCORING KEY AND RATING GUIDE Mechanics of Rating The following procedures are to be followed f scing student answer papers f the Regents Examination in Geometry. Me detailed infmation about scing is provided in the publication Infmation Booklet f Scing the Regents Examinations in Mathematics. Use only red ink red pencil in rating Regents papers. Do not attempt to crect the student s wk by making insertions changes of any kind. Use check marks to indicate student errs. Unless otherwise specified, mathematically crect variations in the answers will be allowed. Units need not be given when the wding of the questions allows such omissions. Each student s answer paper is to be sced by a minimum of three mathematics teachers. On the back of the student s detachable answer sheet, raters must enter their initials in the boxes next to the questions they have sced and also write their name in the box under the heading Rater s/scer s Name. Raters should recd the student s sces f all questions and the total raw sce on the student s detachable answer sheet. Then the student s total raw sce should be converted to a scale sce by using the conversion chart that will be posted on the Department s web site on Thursday, January 27, The student s scale sce should be entered in the box provided on the student s detachable answer sheet. The scale sce is the student s final examination sce.

34 Part I Allow a total of 56 credits, 2 credits f each of the following. Allow credit if the student has written the crect answer instead of the numeral 1, 2, 3, 4. (1) 3 (8) 4 (15) 4 (22) 1 (2) 1 (9) 2 (16) 3 (23) 4 (3) 2 (10) 3 (17) 2 (24) 4 (4) 3 (11) 3 (18) 4 (25) 2 (5) 3 (12) 1 (19) 4 (26) 3 (6) 2 (13) 3 (20) 1 (27) 3 (7) 3 (14) 4 (21) 4 (28) 1 Geometry Jan. 11 [2]

36 Part II F each question, use the specific criteria to award a maximum of two credits. Unless otherwise specified, mathematically crect alternative solutions should be awarded appropriate credit. (29) [2] 30, and appropriate wk is shown, such as crectly labeling angle measures on the diagram. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] 30, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. (30) [2] A ( 2, 2), B ( 1,3), and [1] One graphing err is made. C (4,0). [1] One conceptual err is made, such as reflecting the triangle over the y-axis. [1] Only the codinates of two points are stated and labeled crectly. [1] A, B, and C are graphed and labeled crectly, but the codinates are not stated are stated increctly. [1] ( 2, 2), ( 1,3), and (4,0), but the codinates are not labeled are labeled increctly. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. Geometry Jan. 11 [4]

37 (31) [2] An interi angle of 108 and an exteri angle of 72, and appropriate wk is shown. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] Appropriate wk is shown to find 108 and 72, but the angles are not labeled are labeled increctly. [1] Appropriate wk is shown to find either an interi angle of 108 an exteri angle of 72. [1] An interi angle of 108 and an exteri angle of 72, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. (32) [2] 6 2, and appropriate wk is shown. [1] Appropriate wk is shown, but one computational simplification err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] Appropriate wk is shown, but the answer is not written in simplest radical fm. [1] Appropriate wk is shown, but the answer is written as a decimal. [1] 6 2, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. Geometry Jan. 11 [5]

38 (33) [2] A crect construction is drawn showing all appropriate arcs, and the angle bisect is drawn. [1] All appropriate arcs are drawn, but the angle bisect is not drawn. [1] Appropriate wk is shown, but one construction err is made. [0] A drawing that is not an appropriate construction is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. (34) [2] 1, and appropriate wk is shown. 3 [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] 1, but no wk is shown. 3 [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. Geometry Jan. 11 [6]

39 Part III F each question, use the specific criteria to award a maximum of four credits. Unless otherwise specified, mathematically crect alternative solutions should be awarded appropriate credit. (35) [4] Both loci are graphed crectly and (2,5), (2, 3), and (6,1) are stated. [3] Appropriate wk is shown, but one graphing err is made, but appropriate codinates are stated. [3] Appropriate wk is shown, but the codinates of only one two of the points are stated crectly. [3] Appropriate wk is shown and the crect points are indicated, but the codinates are not stated are stated increctly. [2] Both loci are graphed crectly, but no further crect wk is shown. [2] Appropriate wk is shown, but one graphing err is made, but appropriate points are indicated, but the codinates are not stated are stated increctly. [2] Appropriate wk is shown, but one conceptual err is made, but appropriate codinates are stated. [1] Only one locus is graphed crectly, but no further crect wk is shown. [1] (2,5), (2, 3), and (6,1), but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. Geometry Jan. 11 [7]

40 (36) [4] A complete and crect proof that includes a concluding statement is written. [3] A proof is written that demonstrates a though understanding of the method of proof and contains no conceptual errs, but one statement reason is missing is increct no concluding statement is written. [2] DFE ACB is proven, but no further crect wk is shown. [2] A proof is written that demonstrates a good understanding of the method of proof and contains no conceptual errs, but two statements reasons are missing are increct. [2] A proof is written that demonstrates a good understanding of the method of proof, but one conceptual err is made. [1] ABC DEF is proven, but no further crect wk is shown. [1] Some crect relevant statements about the proof are made, but three four statements reasons are missing are increct. [0] The given and/ the prove statements are written, but no further crect relevant statements are written. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. Geometry Jan. 11 [8]

41 (37) [4] 32, and appropriate wk is shown. [3] Appropriate wk is shown, but one computational err is made. [3] A crect proption is written and x = 35 is found, but no further crect wk is shown. [2] Appropriate wk is shown, but two me computational errs are made. [2] Appropriate wk is shown, but one conceptual err is made. [2] A crect proption is written, but no further crect wk is shown. [1] Appropriate wk is shown, but one conceptual err and one computational err are made. [1] 32, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. Geometry Jan. 11 [9]

42 Part IV F this question, use the specific criteria to award a maximum of six credits. Unless otherwise specified, mathematically crect alternative solutions should be awarded appropriate credit. (38) [6] A complete and crect proof that includes concluding statements that MATH is a rhombus and MATH is not a square is written. [5] Appropriate wk is shown, but one computational graphing err is made. Geometry Jan. 11 [10] [5] Appropriate wk is shown to prove MATH is a rhombus, and wk is shown to prove MATH is not a square, but a concluding statement is missing is increct. [4] Appropriate wk is shown, but two me computational graphing errs are made. [4] Appropriate wk is shown, but one conceptual err is made. [4] Appropriate wk is shown to prove MATH is a rhombus, but no further crect wk is shown. [4] Appropriate wk is shown to prove MATH is a parallelogram and not a square, but no wk is shown to prove it is a rhombus. [3] Appropriate wk is shown, but one conceptual err and one computational graphing err are made. [3] Appropriate wk is shown to prove MATH is a parallelogram, but no further crect wk is shown. [2] Appropriate wk is shown, but two conceptual errs are made. [2] Appropriate wk is shown to find all four crect slopes all four crect sides, but no further crect wk is shown. [2] Appropriate wk is shown to prove the diagonals are perpendicular bisects of each other, but no further crect wk is shown. [2] Appropriate wk is shown to prove MATH is not a square, but no further crect wk is shown.

43 [1] The crect slopes of all four sides are stated the crect lengths of all four sides are stated, but no wk is shown and no proof is written. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. Geometry Jan. 11 [11]

44 Map to Ce Curriculum Content Band Item Numbers Geometric Relationships 5, 9, 17, 28 Constructions 20, 33 Locus 10, 35 Infmal and Fmal Proofs 1, 3, 4, 8, 11, 12, 13, 15, 18, 22, 23, 24, 27, 29, 31, 32, 36, 37 Transfmational Geometry 2, 7, 26, 30 Codinate Geometry 6, 14, 16, 19, 21, 25, 34, 38 Regents Examination in Geometry January 2011 Chart f Converting Total Test Raw Sces to Final Examination Sces (Scale Sces) The Chart f Determining the Final Examination Sce f the January 2011 Regents Examination in Geometry will be posted on the Department s web site on Thursday, January 27, Conversion charts provided f previous administrations of the Geometry examination must NOT be used to determine students final sces f this administration. Online Submission of Teacher Evaluations of the Test to the Department Suggestions and feedback from teachers provide an imptant contribution to the test development process. The Department provides an online evaluation fm f State assessments. It contains spaces f teachers to respond to several specific questions and to make suggestions. Instructions f completing the evaluation fm are as follows: 1. Go to 2. Select the test title. 3. Complete the required demographic fields. 4. Complete each evaluation question and provide comments in the space provided. 5. Click the SUBMIT button at the bottom of the page to submit the completed fm. Geometry Jan. 11 [12]

45 Regents Examination in Geometry January 2011 Chart f Converting Total Test Raw Sces to Final Examination Sces (Scale Sces) Raw Sce Scale Sce Raw Sce Scale Sce Raw Sce Scale Sce Raw Sce Scale Sce To determine the student s final examination sce, find the student s total test raw sce in the column labeled Raw Sce and then locate the scale sce that cresponds to that raw sce. The scale sce is the student s final examination sce. Enter this sce in the space labeled Scale Sce on the student s answer sheet. All student answer papers that receive a scale sce of 60 through 64 must be sced a second time to ensure the accuracy of the sce. F the second scing, a different committee of teachers may sce the student s paper the iginal committee may sce the paper, except that no teacher may sce the same open-ended questions that he/she sced in the first rating of the paper. Because scale sces cresponding to raw sces in the conversion chart change from one examination to another, it is crucial that f each administration, the conversion chart provided f that administration be used to determine the student s final sce. The chart above is usable only f this administration of the Regents Examination in Geometry.

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

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