# 0612ge. Geometry Regents Exam

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1 0612ge 1 Triangle ABC is graphed on the set of axes below. 3 As shown in the diagram below, EF intersects planes P, Q, and R. Which transformation produces an image that is similar to, but not congruent to, ABC? 1) T 2, 3 2) D 2 3) r y = x 4) R 90 2 A student wrote the sentence 4 is an odd integer. What is the negation of this sentence and the truth value of the negation? 1) 3 is an odd integer; true 2) 4 is not an odd integer; true 3) 4 is not an even integer; false 4) 4 is an even integer; false If EF is perpendicular to planes P and R, which statement must be true? 1) Plane P is perpendicular to plane Q. 2) Plane R is perpendicular to plane P. 3) Plane P is parallel to plane Q. 4) Plane R is parallel to plane P. 4 In the diagram below, LATE is an isosceles trapezoid with LE AT, LA = 24, ET = 40, and AT = 10. Altitudes LF and AG are drawn. What is the length of LF? 1) 6 2) 8 3) 3 4) 4 1

2 5 In the diagram below of circle O, diameter AB is parallel to chord CD. 8 The diagram below shows the construction of AB. through point P parallel to CD If mcd = 70, what is mac? 1) 110 2) 70 3) 55 4) 35 6 In the diagram below of ABCD, AC BD. Using this information, it could be proven that 1) BC = AB 2) AB = CD 3) AD BC = CD 4) AB + CD = AD 7 The diameter of a sphere is 15 inches. What is the volume of the sphere, to the nearest tenth of a cubic inch? 1) ) ) ) 14,137.2 Which theorem justifies this method of construction? 1) If two lines in a plane are perpendicular to a transversal at different points, then the lines are parallel. 2) If two lines in a plane are cut by a transversal to form congruent corresponding angles, then the lines are parallel. 3) If two lines in a plane are cut by a transversal to form congruent alternate interior angles, then the lines are parallel. 4) If two lines in a plane are cut by a transversal to form congruent alternate exterior angles, then the lines are parallel. 9 Parallelogram ABCD has coordinates A(1, 5), B(6, 3), C(3, 1), and D( 2, 1). What are the coordinates of E, the intersection of diagonals AC and BD? 1) (2, 2) 2) (4.5, 1) 3) (3.5, 2) 4) ( 1, 3) 10 What is the equation of a circle whose center is 4 units above the origin in the coordinate plane and whose radius is 6? 1) x 2 + (y 6) 2 = 16 2) (x 6) 2 + y 2 = 16 3) x 2 + (y 4) 2 = 36 4) (x 4) 2 + y 2 = 36 2

3 11 In the diagram of ABC shown below, D is the midpoint of AB, E is the midpoint of BC, and F is the midpoint of AC. 14 In the diagram below of ABC, AE BE, AF CF, and CD BD. If AB = 20, BC = 12, and AC = 16, what is the perimeter of trapezoid ABEF? 1) 24 2) 36 3) 40 4) In the diagram below, LMO is isosceles with LO = MO. Point P must be the 1) centroid 2) circumcenter 3) incenter 4) orthocenter 15 What is the equation of the line that passes through the point ( 9, 6) and is perpendicular to the line y = 3x 5? 1) y = 3x ) y = 1 3 x 3 3) y = 3x ) y = 1 3 x + 3 If m L = 55 and m NOM = 28, what is m N? 1) 27 2) 28 3) 42 4) If AB is contained in plane P, and AB is perpendicular to plane R, which statement is true? 1) AB is parallel to plane R. 2) Plane P is parallel to plane R. 3) AB is perpendicular to plane P. 4) Plane P is perpendicular to plane R. 16 In the diagram of ABC shown below, DE BC. If AB = 10, AD = 8, and AE = 12, what is the length of EC? 1) 6 2) 2 3) 3 4) 15 3

4 17 What is the length of AB with endpoints A( 1, 0) and B(4, 3)? 1) 6 2) 18 3) 34 4) The sum of the interior angles of a polygon of n sides is 1) ) n 3) (n 2) 180 (n 2) 180 4) n 19 What is the slope of a line perpendicular to the line whose equation is 20x 2y = 6? 1) Which graph represents a circle whose equation is (x + 2) 2 + y 2 = 16? 1) 2) 2) ) ) 10 3) 4) 4

5 21 In circle O shown below, diameter DB is perpendicular to chord AC at E. 24 In the diagram below, ABC RST. If DB = 34, AC = 30, and DE > BE, what is the length of BE? 1) 8 2) 9 3) 16 4) 25 Which statement is not true? 1) A R AB 2) RS = BC ST AB 3) BC = ST RS AB + BC + AC 4) RS + ST + RT = AB RS 25 In the diagram below of ABC, BC is extended to D. 22 In parallelogram ABCD shown below, diagonals AC and BD intersect at E. Which statement must be true? 1) AC DB 2) ABD CBD 3) AED CEB 4) DCE BCE 23 Which equation of a circle will have a graph that lies entirely in the first quadrant? 1) (x 4) 2 + (y 5) 2 = 9 2) (x + 4) 2 + (y + 5) 2 = 9 3) (x + 4) 2 + (y + 5) 2 = 25 4) (x 5) 2 + (y 4) 2 = 25 If m A = x 2 6x, m B = 2x 3, and m ACD = 9x + 27, what is the value of x? 1) 10 2) 2 3) 3 4) An equation of the line that passes through (2, 1) and is parallel to the line 2y + 3x = 8 is 1) y = 3 2 x 4 2) y = 3 2 x + 4 3) y = 3 2 x 2 4) y = 3 2 x + 2 5

6 27 The graph below shows JT and its image, J T, after a transformation. 29 Triangle TAP has coordinates T( 1, 4), A(2, 4), and P(2, 0). On the set of axes below, graph and label T A P, the image of TAP after the translation (x, y) (x 5, y 1). Which transformation would map JT onto J T? 1) translation 2) glide reflection 3) rotation centered at the origin 4) reflection through the origin 28 Which reason could be used to prove that a parallelogram is a rhombus? 1) Diagonals are congruent. 2) Opposite sides are parallel. 3) Diagonals are perpendicular. 4) Opposite angles are congruent. 30 In the diagram below, m and QR ST at R. If m 1 = 63, find m Two lines are represented by the equations x + 2y = 4 and 4y 2x = 12. Determine whether these lines are parallel, perpendicular, or neither. Justify your answer. 6

7 32 Using a compass and straightedge, construct the bisector of CBA. [Leave all construction marks.] 34 On the set of axes below, graph the locus of points that are 4 units from the line x = 3 and the locus of points that are 5 units from the point (0, 2). Label with an X all points that satisfy both conditions. 33 The cylindrical tank shown in the diagram below is to be painted. The tank is open at the top, and the bottom does not need to be painted. Only the outside needs to be painted. Each can of paint covers 600 square feet. How many cans of paint must be purchased to complete the job? 35 Given: AD bisects BC at E. AB BC DC BC Prove: AB DC 7

8 36 The coordinates of trapezoid ABCD are A( 4, 5), B(1, 5), C(1, 2), and D( 6, 2). Trapezoid A B C D is the image after the composition r x axis r y = x is performed on trapezoid ABCD. State the coordinates of trapezoid A B C D. [The use of the set of axes below is optional.] 37 In the diagram below of circle O, chords RT and QS intersect at M. Secant PTR and tangent PS are drawn to circle O. The length of RM is two more than the length of TM, QM = 2, SM = 12, and PT = 8. Find the length of RT. Find the length of PS. 38 On the set of axes below, solve the system of equations graphically and state the coordinates of all points in the solution. y = (x 2) 2 3 2y + 16 = 4x 8

9 ID: A 0612ge Answer Section 1 ANS: 2 PTS: 2 REF: ge STA: G.G.59 TOP: Properties of Transformations 2 ANS: 2 PTS: 2 REF: ge STA: G.G.24 TOP: Negations 3 ANS: 4 PTS: 2 REF: ge STA: G.G.9 TOP: Planes 4 ANS: = = 6. PTS: 2 REF: ge STA: G.G.40 TOP: Trapezoids 5 ANS: = 55 PTS: 2 REF: ge STA: G.G.52 TOP: Chords 6 ANS: 2 AC = BD AC BC = BD BC AB = CD PTS: 2 REF: ge STA: G.G.27 TOP: Line Proofs 7 ANS: 2 V = 4 3 πr3 = π PTS: 2 REF: ge STA: G.G.16 TOP: Volume and Surface Area 8 ANS: 2 PTS: 2 REF: ge STA: G.G.19 TOP: Constructions 9 ANS: The diagonals of a parallelogram intersect at their midpoints. M AC 2, 5 + ( 1) 2 = (2, 2) PTS: 2 REF: ge STA: G.G.69 TOP: Quadrilaterals in the Coordinate Plane 10 ANS: 3 PTS: 2 REF: ge STA: G.G.71 TOP: Equations of Circles 1

10 ID: A 11 ANS: = 44. PTS: 2 REF: ge STA: G.G.42 TOP: Midsegments 12 ANS: 1 PTS: 2 REF: ge STA: G.G.31 TOP: Isosceles Triangle Theorem 13 ANS: 4 PTS: 2 REF: ge STA: G.G.5 TOP: Planes 14 ANS: 1 PTS: 2 REF: ge STA: G.G.21 TOP: Centroid, Orthocenter, Incenter and Circumcenter 15 ANS: 4 m = 1 3. y = mx + b 6 = 1 3 ( 9) + b 6 = 3 + b 3 = b PTS: 2 REF: ge STA: G.G.64 TOP: Parallel and Perpendicular Lines 16 ANS: = 12 x 8x = 24 x = 3. PTS: 2 REF: ge STA: G.G.46 TOP: Side Splitter Theorem 2

11 ID: A 17 ANS: 3 d = ( 1 4) 2 + (0 ( 3)) 2 = = 34 PTS: 2 REF: ge STA: G.G.67 TOP: Distance KEY: general 18 ANS: 3 PTS: 2 REF: ge STA: G.G.36 TOP: Interior and Exterior Angles of Polygons 19 ANS: 2 m = A B = 20 2 = 10. m = 1 10 PTS: 2 REF: ge STA: G.G.62 TOP: Parallel and Perpendicular Lines 20 ANS: 3 PTS: 2 REF: ge STA: G.G.74 TOP: Graphing Circles 21 ANS: = = 9 PTS: 2 REF: ge STA: G.G.49 TOP: Chords 22 ANS: 3 bisect each other.. Opposite sides of a parallelogram are congruent and the diagonals of a parallelogram PTS: 2 REF: ge STA: G.G.28 TOP: Triangle Congruency 23 ANS: 1 PTS: 2 REF: ge STA: G.G.73 TOP: Equations of Circles 24 ANS: 3 PTS: 2 REF: ge STA: G.G.45 TOP: Similarity KEY: basic 25 ANS: 4 x 2 6x + 2x 3 = 9x + 27 x 2 4x 3 = 9x + 27 x 2 13x 30 = 0 (x 15)(x + 2) = 0 x = 15, 2 PTS: 2 REF: ge STA: G.G.32 TOP: Exterior Angle Theorem 3

12 ID: A 26 ANS: 4 m = A B = 3 2. y = mx + b 3 1 = 2 (2) + b 1 = 3 + b 2 = b PTS: 2 REF: ge STA: G.G.65 TOP: Parallel and Perpendicular Lines 27 ANS: 2 PTS: 2 REF: ge STA: G.G.56 TOP: Identifying Transformations 28 ANS: 3 PTS: 2 REF: ge STA: G.G.39 TOP: Special Parallelograms 29 ANS: T ( 6, 3), A ( 3, 3), P ( 3, 1) PTS: 2 REF: ge STA: G.G.61 TOP: Analytical Representations of Transformations 30 ANS: 180 ( ) = 27 PTS: 2 REF: ge STA: G.G.35 TOP: Parallel Lines and Transversals 31 ANS: The slope of x + 2y = 4 is m = A B = 1 A. The slope of 4y 2x = 12 is 2 B = 2 4 = 1. Since the slopes are neither 2 equal nor opposite reciprocals, the lines are neither parallel nor perpendicular. PTS: 2 REF: ge STA: G.G.63 TOP: Parallel and Perpendicular Lines 32 ANS: PTS: 2 REF: ge STA: G.G.17 TOP: Constructions 4

13 ID: A 33 ANS: L = 2πrh = 2π cans are needed. PTS: 2 REF: ge STA: G.G.14 TOP: Volume and Lateral Area 34 ANS: PTS: 2 REF: ge STA: G.G.23 TOP: Locus 35 ANS: B and C are right angles because perpendicular lines form right angles. B C because all right angles are congruent. AEB DEC because vertical angles are congruent. ABE DCE because of ASA. AB DC because CPCTC. PTS: 4 REF: ge STA: G.G.27 TOP: Triangle Proofs 36 ANS: A (5, 4), B (5, 1), C (2, 1), D (2, 6); A (5, 4), B (5, 1), C (2, 1), D (2, 6) PTS: 4 REF: ge STA: G.G.58 TOP: Compositions of Transformations KEY: grids 5

14 ID: A 37 ANS: x(x + 2) = RT = = 10. y y = 18 8 x 2 + 2x 24 = 0 (x + 6)(x 4) = 0 x = 4 y 2 = 144 y = 12 PTS: 4 REF: ge STA: G.G.53 TOP: Segments Intercepted by Circle KEY: tangent and secant 38 ANS: PTS: 6 REF: ge STA: G.G.70 TOP: Quadratic-Linear Systems 6

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