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

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3 3 In the diagram below, line m is parallel to line n. Figure 2 is the image of Figure 1 after a reflection over line m. Figure 3 is the image of Figure 2 after a reflection over line n. Use this space for computations. Figure 1 m Figure 2 Figure 3 n Which single transformation would carry Figure 1 onto Figure 3? (1) a dilation (3) a reflection (2) a rotation (4) a translation 4 In the diagram below, AF and DB intersect at C, and AD and FBE are drawn such that m D 65, m CBE 115, DC 7.2, AC 9.6, and FC F 21.6 D A C E B What is the length of CB? (1) 3.2 (3) 16.2 (2) 4.8 (4) 19.2 Geometry June 18 [3] [OVER]

4 5 Given square RSTV, where RS 9 cm. If square RSTV is dilated by a scale factor of 3 about a given center, what is the perimeter, in centimeters, of the image of RSTV after the dilation? (1) 12 (3) 36 (2) 27 (4) 108 Use this space for computations. 6 In right triangle ABC, hypotenuse AB has a length of 26 cm, and side BC has a length of 17.6 cm. What is the measure of angle B, to the nearest degree? (1) 48 (3) 43 (2) 47 (4) 34 7 The greenhouse pictured below can be modeled as a rectangular prism with a half-cylinder on top. The rectangular prism is 20 feet wide, 12 feet high, and 45 feet long. The half-cylinder has a diameter of 20 feet. 12 ft 20 ft 45 ft To the nearest cubic foot, what is the volume of the greenhouse? (1) 17,869 (3) 39,074 (2) 24,937 (4) 67,349 Geometry June 18 [4]

5 8 In a right triangle, the acute angles have the relationship sin (2x + 4) cos (46). Use this space for computations. What is the value of x? (1) 20 (3) 24 (2) 21 (4) 25 9 In the diagram below, AB DFC, EDA CBG, and EFB and AG are drawn. E D F C A B G Which statement is always true? (1) DEF CBF (3) BAG ~ AEB (2) BAG BAE (4) DEF ~ AEB 10 The base of a pyramid is a rectangle with a width of 4.6 cm and a length of 9 cm. What is the height, in centimeters, of the pyramid if its volume is 82.8 cm 3? (1) 6 (3) 9 (2) 2 (4) 18 Geometry June 18 [5] [OVER]

6 11 In the diagram below of right triangle AED, BC DE. Use this space for computations. D B A Which statement is always true? (1) AC DE BC AE (3) (2) AB BC AD DE (4) C AC CE DE BC BC DE DB AB E 12 What is an equation of the line that passes through the point (6,8) 3 and is perpendicular to a line with equation y x 5? (1) y 8 (x 6) (3) y 8 (x 6) (2) y 8 (x 6) (4) y 8 (x 6) 3 3 Geometry June 18 [6]

7 13 The diagram below shows parallelogram ABCD with diagonals and BD intersecting at E. AC Use this space for computations. B C E A D What additional information is sufficient to prove that parallelogram ABCD is also a rhombus? (1) BD bisects AC. (3) AC is congruent to BD. (2) AB is parallel to CD. (4) AC is perpendicular to BD. 14 Directed line segment DE has endpoints D( 4, 2) and E(1,8). Point F divides DE such that DF:FE is 2:3. What are the coordinates of F? (1) ( 3,0) (3) ( 1,4) (2) ( 2,2) (4) (2,4) Geometry June 18 [7] [OVER]

8 15 Triangle DAN is graphed on the set of axes below. The vertices of DAN have coordinates D( 6, 1), A(6,3), and N( 3,10). Use this space for computations. N y A x D What is the area of DAN? (1) 60 (3) (2) 120 (4) Geometry June 18 [8]

9 16 Triangle ABC, with vertices at A(0,0), B(3,5), and C(0,5), is graphed on the set of axes shown below. Use this space for computations. y C B A x Which figure is formed when ABC is rotated continuously about BC? (1) (3) (2) (4) Geometry June 18 [9] [OVER]

10 17 In the diagram below of circle O, chords AB and CD intersect at E. Use this space for computations. A D E C O B If m AC 72 and m AEC 58, how many degrees are in m DB? (1) 108 (3) 44 (2) 65 (4) In triangle SRK below, medians SC, KE, and RL intersect at M. S E M L R C K Which statement must always be true? (1) 3(MC) SC (3) RM 2MC 1 (2) MC (SM) (4) SM KM 3 Geometry June 18 [10]

11 19 The regular polygon below is rotated about its center. Use this space for computations. Which angle of rotation will carry the figure onto itself? (1) 60 (3) 216 (2) 108 (4) What is an equation of circle O shown in the graph below? y O x (1) x 2 10x y 2 4y 13 (3) x 2 10x y 2 4y 25 (2) x 2 10x y 2 4y 13 (4) x 2 10x y 2 4y 25 Geometry June 18 [11] [OVER]

12 21 In the diagram below of PQR, ST is drawn parallel to PR, PS 2, SQ 5, and TR 5. Use this space for computations. Q 5 2 P S T 5 R What is the length of QR? (1) 7 (3) (2) 2 (4) The diagram below shows circle O with radii OA and OB. The measure of angle AOB is 120, and the length of a radius is 6 inches. A O 6 B Which expression represents the length of arc AB, in inches? 120 (1) (3) 360 ( 6π) 1 3 (36π) (2) 120(6) (4) 1 3 (12π) Geometry June 18 [12]

13 23 Line segment CD is the altitude drawn to hypotenuse EF in right triangle ECF. If EC 10 and EF 24, then, to the nearest tenth, ED is (1) 4.2 (3) 15.5 (2) 5.4 (4) 21.8 Use this space for computations. 24 Line MN is dilated by a scale factor of 2 centered at the point (0,6). If MN is represented by y 3x 6, which equation can represent MN, the image of MN? (1) y 3x 12 (3) y 6x 12 (2) y 3x 6 (4) y 6x 6 Geometry June 18 [13] [OVER]

14 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 Triangle A B C is the image of triangle ABC after a translation of 2 units to the right and 3 units up. Is triangle ABC congruent to triangle A B C? Explain why. Geometry June 18 [14]

15 26 Triangle ABC and point D(1,2) are graphed on the set of axes below. y B A D C x Graph and label A B C, the image of ABC, after a dilation of scale factor 2 centered at point D. Geometry June 18 [15] [OVER]

16 27 Quadrilaterals BIKE and GOLF are graphed on the set of axes below. y F L G x E B K O I Describe a sequence of transformations that maps quadrilateral BIKE onto quadrilateral GOLF. Geometry June 18 [16]

17 28 In the diagram below, secants RST and RQP, drawn from point R, intersect circle O at S, T, Q, and P. T S R Q O P If RS 6, ST 4, and RP 15, what is the length of RQ? Geometry June 18 [17] [OVER]

18 29 Using a compass and straightedge, construct the median to side AC in ABC below. [Leave all construction marks.] A B C Geometry June 18 [18]

19 30 Skye says that the two triangles below are congruent. Margaret says that the two triangles are similar. 5 cm 12 cm 5 cm 13 cm Are Skye and Margaret both correct? Explain why. Geometry June 18 [19] [OVER]

20 31 Randy s basketball is in the shape of a sphere with a maximum circumference of 29.5 inches. Determine and state the volume of the basketball, to the nearest cubic inch. Geometry June 18 [20]

21 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 Triangle ABC has vertices with coordinates A( 1, 1), B(4,0), and C(0,4). Prove that ABC is an isosceles triangle but not an equilateral triangle. [The use of the set of axes below is optional.] y x Geometry June 18 [21] [OVER]

22 33 The map of a campground is shown below. Campsite C, first aid station F, and supply station S lie along a straight path. The path from the supply station to the tower, T, is perpendicular to the path from the supply station to the campsite. The length of path FS is 400 feet. The angle formed by path TF and path FS is 72. The angle formed by path TC and path CS is 55. Campground Map Tower T C F 400 ft S Campsite First aid Supply station Determine and state, to the nearest foot, the distance from the campsite to the tower. Geometry June 18 [22]

23 34 Shae has recently begun kickboxing and purchased training equipment as modeled in the diagram below. The total weight of the bag, pole, and unfilled base is 270 pounds. The cylindrical base is 18 inches tall with a diameter of 20 inches. The dry sand used to fill the base weighs lbs per cubic foot. Bag 18 in Base 20 in To the nearest pound, determine and state the total weight of the training equipment if the base is filled to 85% of its capacity. Geometry June 18 [23] [OVER]

24 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. Utilize the information provided for the question to determine your answer. Note that diagrams are not necessarily drawn to scale. 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. [6] 35 Given: Parallelogram ABCD, BF AFD, and DE BEC B E C A F D Prove: BEDF is a rectangle Work space for question 35 is continued on the next page. Geometry June 18 [24]

25 Question 35 continued Geometry June 18 [25]

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

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

31 High School Math Reference Sheet Tear Here 1 inch 2.54 centimeters 1 kilometer 0.62 mile 1 cup 8 fluid ounces 1 meter inches 1 pound 16 ounces 1 pint 2 cups 1 mile 5280 feet 1 pound kilogram 1 quart 2 pints 1 mile 1760 yards 1 kilogram 2.2 pounds 1 gallon 4 quarts 1 mile kilometers 1 ton 2000 pounds 1 gallon liters 1 liter gallon 1 liter 1000 cubic centimeters 1 Triangle A bh 2 Pythagorean Theorem a 2 b 2 c 2 Parallelogram A bh Quadratic Formula x b 4ac 2a b 2 Circle A πr 2 Arithmetic Sequence a n a 1 (n 1)d Circle C πd or C 2πr Geometric Sequence a n a 1 r n 1 General Prisms V Bh Geometric Series n a a r 1 1 S n where r 1 1 r Cylinder V πr 2 h 180 Radians 1 radian degrees π 4 Sphere V πr 3 3 π Degrees 1 degree radians Cone V πr 2 h 3 1 Pyramid V Bh 3 Exponential Growth/Decay A A 0 e k(t t 0) B 0 Tear Here Geometry June 18 [31]

32 GEOMETRY Tear Here Tear Here Printed on Recycled Paper GEOMETRY

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

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