# The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

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2 Part I nswer all 28 questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. For each question, write on the separate answer sheet the numeral preceding the word or expression that best completes the statement or answers the question. [56] 1 Line n intersects lines l and m, forming the angles shown in the diagram below. Use this space for computations. n (6x + 42) l (18x 12) m Which value of x would prove l m? (1) 2.5 (3) 6.25 (2) 4.5 (4) In a given triangle, the point of intersection of the three medians is the same as the point of intersection of the three altitudes. Which classification of the triangle is correct? (1) scalene triangle (3) equilateral triangle (2) isosceles triangle (4) right isosceles triangle 3 circle has the equation (x 2) 2 (y 3) What are the coordinates of its center and the length of its radius? (1) ( 2,3) and 6 (3) ( 2,3) and 36 (2) (2, 3) and 6 (4) (2, 3) and 36 Geometry January 12 [2]

3 4 In the diagram below, MTH is a rhombus with diagonals H and MT. Use this space for computations. 12 M T H If m HM 12, what is m MT? (1) 12 (3) 84 (2) 78 (4) line segment has endpoints (4,7) and (1,11). What is the length of the segment? (1) 5 (3) 16 (2) 7 (4) 25 6 In FGH, m F 42 and an exterior angle at vertex H has a measure of 104. What is m G? (1) 34 (3) 76 (2) 62 (4) 146 Geometry January 12 [3] [OVER]

4 7 Which diagram represents a correct construction of equilateral BC, given side B? Use this space for computations. C C B B (1) (3) C C B B (2) (4) 8 In the diagram below, BC is circumscribed about circle O and the sides of BC are tangent to the circle at points D, E, and F. B E O F If B 20, E 12, and CF 15, what is the length of C? (1) 8 (3) 23 (2) 15 (4) 27 D C Geometry January 12 [4]

5 C 9 In BC and DEF, CB. Which additional information DF FE would prove BC DEF? Use this space for computations. (1) C DF (3) CB DFE (2) CB FE (4) BC EDF 10 The angles of triangle BC are in the ratio of 8:3:4. What is the measure of the smallest angle? (1) 12 (3) 36 (2) 24 (4) When a quadrilateral is reflected over the line y x, which geometric relationship is not preserved? (1) congruence (3) parallelism (2) orientation (4) perpendicularity 12 Which equation represents circle O with center (2, 8) and radius 9? (1) (x 2) 2 (y 8) 2 9 (2) (x 2) 2 (y 8) 2 9 (3) (x 2) 2 (y 8) 2 81 (4) (x 2) 2 (y 8) 2 81 Geometry January 12 [5] [OVER]

6 13 Which statement is the negation of Two is a prime number and what is the truth value of the negation? (1) Two is not a prime number; false (2) Two is not a prime number; true (3) prime number is two; false (4) prime number is two; true Use this space for computations. 14 In the diagram below of circle O, chords B and CD intersect at E. 50 C 34 O E B D If m EC 34 and m C 50, what is m DB? (1) 16 (3) 68 (2) 18 (4) The volume of a rectangular prism is 144 cubic inches. The height of the prism is 8 inches. Which measurements, in inches, could be the dimensions of the base? (1) 3.3 by 5.5 (3) 12 by 8 (2) 2.5 by 7.2 (4) 9 by 9 Geometry January 12 [6]

7 16 The diagram below shows a pair of congruent triangles, with DB CDB and BD CBD. Use this space for computations. B C Which statement must be true? (1) DB CBD (3) B CD (2) BC DC (4) D CD D 17 What is an equation of the line that is perpendicular to the line whose equation is y 3 5 x 2 and that passes through the point (3, 6)? (1) y 5 3 x 11 (3) y 5 3 x 1 (2) y 5 3 x 11 (4) y 5 3 x 1 18 Point lies in plane B. How many lines can be drawn perpendicular to plane B through point? (1) one (3) zero (2) two (4) infinite Geometry January 12 [7] [OVER]

8 19 In the diagram below of isosceles trapezoid BCD, B CD 25, D 26, and BC 12. Use this space for computations. B 12 C D What is the length of an altitude of the trapezoid? (1) 7 (3) 19 (2) 14 (4) What is an equation of circle O shown in the graph below? y O (- 2,2) x (1) (x 2) 2 (y 2) 2 9 (2) (x 2) 2 (y 2) 2 3 (3) (x 2) 2 (y 2) 2 9 (4) (x 2) 2 (y 2) 2 3 Geometry January 12 [8]

9 21 The diagram below represents a rectangular solid. Use this space for computations. E F B H G D C Which statement must be true? (1) EH and BC are coplanar. (2) FG and B are coplanar. (3) EH and D are skew. (4) FG and CG are skew. 22 In RST, m R 58 and m S 73. Which inequality is true? (1) RT TS RS (3) RT RS TS (2) RS RT TS (4) RS TS RT 23 The number of degrees in the sum of the interior angles of a pentagon is (1) 72 (3) 540 (2) 360 (4) What is the equation of a line passing through (2, 1) and parallel to the line represented by the equation y 2x 1? (1) y 1 x (3) y 2x 5 2 (2) y 1 2 x 1 (4) y 2x 1 Geometry January 12 [9] [OVER]

10 25 The coordinates of the endpoints of B are (0,0) and B(0,6). The equation of the perpendicular bisector of B is (1) x 0 (3) y 0 (2) x 3 (4) y 3 Use this space for computations. 26 In the diagram below, point P is the centroid of BC. B N P L If PM 2x 5 and BP 7x 4, what is the length of PM? (1) 9 (3) 18 (2) 2 (4) 27 M C 27 In PQR, PRQ is a right angle and RT is drawn perpendicular to hypotenuse PQ. If PT x, RT 6, and TQ 4x, what is the length of PQ? (1) 9 (3) 3 (2) 12 (4) In BC, B 5 feet and BC 3 feet. Which inequality represents all possible values for the length of C, in feet? (1) 2 C 8 (3) 3 C 7 (2) 2 C 8 (4) 3 C 7 Geometry January 12 [10]

11 Part II nswer all 6 questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. ll answers should be written in pen, except for graphs and drawings, which should be done in pencil. [12] 29 In the diagram below, two parallel lines intersect circle O at points, B, C, and D, with m B x 20 and mdc 2x 20. Find m B. D (x + 20) (2x 20) C O B Geometry January 12 [11] [OVER]

12 30 In the diagram below, point M is located on _ B. Sketch the locus of points that are 1 unit from _ B and the locus of points 2 units from point M. Label with an X all points that satisfy both conditions. M B Geometry January 12 [12]

13 31 Determine whether the two lines represented by the equations y 2x 3 and 2y x 6 are parallel, perpendicular, or neither. Justify your response. Geometry January 12 [13] [OVER]

14 32 The coordinates of the vertices of RST are R( 2,3), S(4,4), and T(2, 2). Triangle R S T is the image of RST after a rotation of 90 about the origin. State the coordinates of the vertices of R S T. [The use of the set of axes below is optional.] y x Geometry January 12 [14]

15 33 On the diagram below, use a compass and straightedge to construct the bisector of XYZ. [Leave all construction marks.] Y X Z Geometry January 12 [15] [OVER]

16 34 In the diagram below of circle O, diameter B is perpendicular to chord CD at E. If O 10 and BE 4, find the length of CE. C B E O D Geometry January 12 [16]

17 Part III nswer 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. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. ll answers should be written in pen, except for graphs and drawings, which should be done in pencil. [12] 35 Triangle BC has coordinates (2, 2), B(2,1), and C(4, 2). Triangle B C is the image of BC under T 5, 2. On the set of axes below, graph and label BC and its image, B C. Determine the relationship between the area of BC and the area of B C. Justify your response. y x Geometry January 12 [17] [OVER]

18 36 paint can is in the shape of a right circular cylinder. The volume of the paint can is 600 cubic inches and its altitude is 12 inches. Find the radius, in inches, of the base of the paint can. Express the answer in simplest radical form. Find, to the nearest tenth of a square inch, the lateral area of the paint can. Geometry January 12 [18]

19 37 Triangle HKL has vertices H( 7,2), K(3, 4), and L(5,4). The midpoint of HL is M and the midpoint of LK is N. Determine and state the coordinates of points M and N. Justify the statement: MN is parallel to HK. [The use of the set of axes below is optional.] y x Geometry January 12 [19] [OVER]

20 Part IV nswer the question in this part. correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. correct numerical answer with no work shown will receive only 1 credit. The answer should be written in pen. [6] 38 In the diagram below of quadrilateral BCD, D BC and DE BCE. Line segments C, DB, and FG intersect at E. Prove: EF CEG D C F E G B Geometry January 12 [20]

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23 Reference Sheet Tear Here Tear Here Volume Lateral rea (L) Surface rea Cylinder Pyramid Right Circular Cone Sphere Right Circular Cylinder Right Circular Cone Sphere V Bh where B is the area of the base V 1 Bh 3 where B is the area of the base V 1 Bh 3 where B is the area of the base V 4 r 3 3 L 2 rh L rl where l is the slant height S 4 r 2 Geometry January 12

24 Tear Here Tear Here

25 Tear Here Tear Here Scrap Graph Paper This sheet will not be scored.

26 Scrap Graph Paper This sheet will not be scored. Tear Here Tear Here

27 Tear Here Tear Here The University of the State of New York REGENTS HIGH SCHOOL EXMINTION GEOMETRY NSWER SHEET Student Sex: Male Female Grade Teacher School Thursday, January 26, :15 a.m. to 12:15 p.m., only Your answers to Part I should be recorded on this answer sheet. Part I nswer all 28 questions in this part Your answers for Parts II, III, and IV should be written in the test booklet. The declaration below must be signed when you have completed the examination. I do hereby affirm, at the close of this examination, that I had no unlawful knowledge of the questions or answers prior to the examination and that I have neither given nor received assistance in answering any of the questions during the examination. Signature Geometry January 12

28 GEOMETRY Rater s/scorer s Name (minimum of three) Question Maximum Credit GEOMETRY Credits Earned Rater s/scorer s Initials Tear Here Part I Part II Part III Part IV 38 6 Maximum Total 86 Total Raw Score Checked by Scale Score (from conversion chart) Tear Here Geometry January 12 Printed on Recycled Paper GEOMETRY

### 1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT.

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