# The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, June 19, :15 a.m. to 12:15 p.m., only.

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2 Part I Answer all 28 questions in this part. Each correct answer will receive 2 credits. For each statement or question, choose the word or expression that, of those given, best completes the statement or answers the question. Record your answers on your separate answer sheet. [56] 1 In trapezoid RSTV below with bases RS and VT, diagonals RT and SV intersect at Q. Use this space for computations. V T Q R S If trapezoid RSTV is not isosceles, which triangle is equal in area to RSV? (1) RQV (3) RVT (2) RST (4) SVT 2 In the diagram below, XYV TSV. W X Y V S T U Which statement can not be proven? (1) XVY TVS (3) XY TS (2) VYX VUT (4) YV SV Geometry June 13 [2]

3 3 In a park, two straight paths intersect. The city wants to install lampposts that are both equidistant from each path and also 15 feet from the intersection of the paths. How many lampposts are needed? (1) 1 (3) 3 (2) 2 (4) 4 Use this space for computations. 4 What are the coordinates of A, the image of point A( 3,4), after a rotation of 180 about the origin? (1) (4, 3) (3) (3,4) (2) ( 4, 3) (4) (3, 4) 5 Based on the construction below, which conclusion is not always true? C A E B D (1) AB CD (3) AE EB (2) AB CD (4) CE DE Geometry June 13 [3] [OVER]

4 6 Which equation represents the circle whose center is ( 5,3) and that passes through the point ( 1,3)? (1) (x 1) 2 (y 3) 2 16 (2) (x 1) 2 (y 3) 2 16 (3) (x 5) 2 (y 3) 2 16 (4) (x 5) 2 (y 3) 2 16 Use this space for computations. 7 As shown in the diagram below, when right triangle DAB is reflected over the x-axis, its image is triangle DCB. y A D B x C Which statement justifies why AB CB? (1) Distance is preserved under reflection. (2) Orientation is preserved under reflection. (3) Points on the line of reflection remain invariant. (4) Right angles remain congruent under reflection. Geometry June 13 [4]

5 8 In ABC, m A 3x 1, m B 4x 17, and m C 5x 20. Which type of triangle is ABC? (1) right (3) isosceles (2) scalene (4) equilateral Use this space for computations. 9 What is the equation for circle O shown in the graph below? y O x (1) (x 3) 2 (y 1) 2 6 (2) (x 3) 2 (y 1) 2 6 (3) (x 3) 2 (y 1) 2 9 (4) (x 3) 2 (y 1) 2 9 Geometry June 13 [5] [OVER]

6 10 Point A is on line m. How many distinct planes will be perpendicular to line m and pass through point A? (1) one (3) zero (2) two (4) infinite Use this space for computations. 11 In ABC, D is the midpoint of AB and E is the midpoint of BC. If AC 3x 15 and DE 6, what is the value of x? (1) 6 (3) 9 (2) 7 (4) What are the coordinates of the center of a circle if the endpoints of its diameter are A(8, 4) and B( 3,2)? (1) (2.5,1) (3) (5.5, 3) (2) (2.5, 1) (4) (5.5,3) Geometry June 13 [6]

7 13 Which graph could be used to find the solution to the following system of equations? Use this space for computations. y (x 3) 2 1 x y 2 (1) (3) (2) (4) 14 What is the converse of If an angle measures 90 degrees, then it is a right angle? (1) If an angle is a right angle, then it measures 90 degrees. (2) An angle is a right angle if it measures 90 degrees. (3) If an angle is not a right angle, then it does not measure 90 degrees. (4) If an angle does not measure 90 degrees, then it is not a right angle. Geometry June 13 [7] [OVER]

8 15 As shown in the diagram below, a right pyramid has a square base, ABCD, and EF is the slant height. Use this space for computations. E D C F A Which statement is not true? (1) EA EC (2) EB EF B (3) AEB BEC (4) CED is isosceles 16 What is the equation of a line passing through the point (6,1) and parallel to the line whose equation is 3x 2y 4? (1) y 2 3 x 5 (3) y 3 2 x 8 (2) y 2 3 x 3 (4) y 3 2 x 5 17 The volume of a sphere is approximately cubic centimeters. What is the radius of the sphere, to the nearest tenth of a centimeter? (1) 2.2 (3) 4.4 (2) 3.3 (4) 4.7 Geometry June 13 [8]

9 18 Points A(5,3) and B(7,6) lie on AB. Points C(6,4) and D(9,0) lie on CD. Which statement is true? (1) AB CD (2) AB CD (3) AB and CD are the same line. (4) AB and CD intersect, but are not perpendicular. Use this space for computations. 19 Which set of equations represents two circles that have the same center? (1) x 2 (y 4) 2 16 and (x 4) 2 y 2 16 (2) (x 3) 2 (y 3) 2 16 and (x 3) 2 (y 3) 2 25 (3) (x 7) 2 (y 2) 2 16 and (x 7) 2 (y 2) 2 25 (4) (x 2) 2 (y 5) 2 16 and (x 2) 2 (y 5) Transversal EF intersects AB and CD, as shown in the diagram below. A E B C F D Which statement could always be used to prove AB CD? (1) 2 4 (3) 3 and 6 are supplementary (2) 7 8 (4) 1 and 5 are supplementary Geometry June 13 [9] [OVER]

10 21 In ABC, m A 60, m B 80, and m C 40. Which inequality is true? (1) AB BC (3) AC BA (2) AC BC (4) BC BA Use this space for computations. 22 Circle O with AOC and ABC is shown in the diagram below. A O C B What is the ratio of m AOC to m ABC? (1) 1:1 (3) 3:1 (2) 2:1 (4) 1:2 23 A rectangular prism has a base with a length of 25, a width of 9, and a height of 12. A second prism has a square base with a side of 15. If the volumes of the two prisms are equal, what is the height of the second prism? (1) 6 (3) 12 (2) 8 (4) In triangles ABC and DEF, AB 4, AC 5, DE 8, DF 10, and A D. Which method could be used to prove ABC DEF? (1) AA (3) SSS (2) SAS (4) ASA Geometry June 13 [10]

11 25 Which graph represents a circle whose equation is x 2 (y 1) 2 9? Use this space for computations. y y x x (1) y (3) y x x (2) (4) 26 What is the perimeter of a rhombus whose diagonals are 16 and 30? (1) 92 (3) 60 (2) 68 (4) 17 Geometry June 13 [11] [OVER]

12 27 In right triangle ABC shown in the diagram below, altitude BD is drawn to hypotenuse AC, CD 12, and AD 3. Use this space for computations. A 3 D 12 B C What is the length of AB? (1) 5 3 (3) 3 5 (2) 6 (4) 9 28 Secants JKL and JMN are drawn to circle O from an external point, J. If JK 8, LK 4, and JM 6, what is the length of JN? (1) 16 (3) 10 (2) 12 (4) 8 Geometry June 13 [12]

13 Part II Answer 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. All answers should be written in pen, except for graphs and drawings, which should be done in pencil. [12] 29 A right circular cylinder has a height of 7 inches and the base has a diameter of 6 inches. Determine the lateral area, in square inches, of the cylinder in terms of. Geometry June 13 [13] [OVER]

14 30 Determine, in degrees, the measure of each interior angle of a regular octagon. Geometry June 13 [14]

15 31 Triangle ABC has vertices at A(3,0), B(9, 5), and C(7, 8). Find the length of AC in simplest radical form. Geometry June 13 [15] [OVER]

16 32 On the ray drawn below, using a compass and straightedge, construct an equilateral triangle with a vertex at R. The length of a side of the triangle must be equal to a length of the diagonal of rectangle ABCD. D C A B R Geometry June 13 [16]

17 33 On the set of axes below, graph the locus of points 4 units from the x-axis and equidistant from the points whose coordinates are ( 2,0) and (8,0). Mark with an X all points that satisfy both conditions. y x Geometry June 13 [17] [OVER]

18 34 The coordinates of two vertices of square ABCD are A(2,1) and B(4,4). Determine the slope of side BC. Geometry June 13 [18]

19 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. 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 The coordinates of the vertices of parallelogram SWAN are S(2, 2), W( 2, 4), A( 4,6), and N(0,8). State and label the coordinates of parallelogram S W A N, the image of SWAN after the transformation T 4, 2 D. [The use of the set of axes below is optional.] 1 2 y x Geometry June 13 [19] [OVER]

20 36 In circle O shown below, chords AB and CD and radius OA are drawn, such that AB CD, OE AB, OF CD, OF 16, CF y 10, and CD 4y 20. D A F O C E B Determine the length of DF. Determine the length of OA. Geometry June 13 [20]

21 37 If RST ABC, m A x 2 8x, m C 4x 5, and m R 5x 30, find m C. [Only an algebraic solution can receive full credit.] Geometry June 13 [21] [OVER]

22 Part IV Answer the question in this part. A correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. A correct numerical answer with no work shown will receive only 1 credit. The answer should be written in pen, except for graphs and drawings, which should be done in pencil. [6] 38 In the diagram of MAH below, MH AH and medians AB and MT are drawn. Prove: MBA ATM M A B T H Geometry June 13 [22]

23 Reference Sheet Tear Here Tear Here Volume Lateral Area (L) Surface Area 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 SA 4 r 2 Geometry June 13

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

28 GEOMETRY Printed on Recycled Paper GEOMETRY

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