# 0112ge. Geometry Regents Exam Line n intersects lines l and m, forming the angles shown in the diagram below.

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1 Geometry Regents Exam ge 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. Which value of x would prove l m? 1).5 ) 4.5 3) 6.5 4) 8.75 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 ) isosceles triangle 3) equilateral triangle 4) right isosceles triangle If m HAM = 1, what is m AMT? 1) 1 ) 78 3) 84 4) A line segment has endpoints (4, 7) and (1, 11). What is the length of the segment? 1) 5 ) 7 3) 16 4) 5 3 A circle has the equation (x ) + (y + 3) = 36. What are the coordinates of its center and the length of its radius? 1) (, 3) and 6 ) (, 3) and 6 3) (, 3) and 36 4) (, 3) and 36 6 In FGH, m F = 4 and an exterior angle at vertex H has a measure of 104. What is m G? 1) 34 ) 6 3) 76 4) 146 1

2 Geometry Regents Exam Which diagram represents a correct construction of equilateral ABC, given side AB? 8 In the diagram below, ABC is circumscribed about circle O and the sides of ABC are tangent to the circle at points D, E, and F. 1) ) 3) If AB = 0, AE = 1, and CF = 15, what is the length of AC? 1) 8 ) 15 3) 3 4) 7 9 In ABC and DEF, AC DF = CB FE. Which additional information would prove ABC DEF? 1) AC = DF ) CB = FE 3) ACB DFE 4) BAC EDF 10 The angles of triangle ABC are in the ratio of 8:3:4. What is the measure of the smallest angle? 1) 1º ) 4º 3) 36º 4) 7º 4) 11 When a quadrilateral is reflected over the line y = x, which geometric relationship is not preserved? 1) congruence ) orientation 3) parallelism 4) perpendicularity

3 Geometry Regents Exam Which equation represents circle O with center (, 8) and radius 9? 1) (x + ) + (y 8) = 9 ) (x ) + (y + 8) = 9 3) (x + ) + (y 8) = 81 4) (x ) + (y + 8) = The diagram below shows a pair of congruent triangles, with ADB CDB and ABD CBD. 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 ) Two is not a prime number; true 3) A prime number is two; false 4) A prime number is two; true 14 In the diagram below of circle O, chords AB and CD intersect at E. Which statement must be true? 1) ADB CBD ) ABC ADC 3) AB CD 4) AD CD 17 What is an equation of the line that is perpendicular to the line whose equation is y = 3 x and that 5 passes through the point (3, 6)? 1) y = 5 3 x 11 ) y = 5 3 x + 11 If m AEC = 34 and mac = 50, what is mdb? 1) 16 ) 18 3) 68 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 ).5 by 7. 3) 1 by 8 4) 9 by 9 3) y = 5 3 x 1 4) y = 5 3 x Point A lies in plane B. How many lines can be drawn perpendicular to plane B through point A? 1) one ) two 3) zero 4) infinite 3

4 Geometry Regents Exam In the diagram below of isosceles trapezoid ABCD, AB = CD = 5, AD = 6, and BC = 1. 1 The diagram below represents a rectangular solid. What is the length of an altitude of the trapezoid? 1) 7 ) 14 3) 19 4) 4 Which statement must be true? 1) EH and BC are coplanar ) FG and AB are coplanar 3) EH and AD are skew 4) FG and CG are skew 0 What is an equation of circle O shown in the graph below? In RST, m R = 58 and m S = 73. Which inequality is true? 1) RT < TS < RS ) RS < RT < TS 3) RT < RS < TS 4) RS < TS < RT 1) (x + ) + (y ) = 9 ) (x + ) + (y ) = 3 3) (x ) + (y + ) = 9 4) (x ) + (y + ) = 3 3 The number of degrees in the sum of the interior angles of a pentagon is 1) 7 ) 360 3) 540 4) 70 4 What is the equation of a line passing through (, 1) and parallel to the line represented by the equation y = x + 1? 1) y = 1 x ) y = 1 x + 1 3) y = x 5 4) y = x 1 4

5 Geometry Regents Exam The coordinates of the endpoints of AB are A(0, 0) and B(0, 6). The equation of the perpendicular bisector of AB is 1) x = 0 ) x = 3 3) y = 0 4) y = 3 9 In the diagram below, two parallel lines intersect circle O at points A, B, C, and D, with mab = x + 0 and mdc = x 0. Find mab. 6 In the diagram below, point P is the centroid of ABC. 30 In the diagram below, point M is located on AB. Sketch the locus of points that are 1 unit from AB and the locus of points units from point M. Label with an X all points that satisfy both conditions. If PM = x + 5 and BP = 7x + 4, what is the length of PM? 1) 9 ) 3) 18 4) 7 7 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 ) 1 3) 3 4) 15 8 In ABC, AB = 5 feet and BC = 3 feet. Which inequality represents all possible values for the length of AC, in feet? 1) AC 8 ) < AC < 8 3) 3 AC 7 4) 3 < AC < 7 31 Determine whether the two lines represented by the equations y = x + 3 and y + x = 6 are parallel, perpendicular, or neither. Justify your response. 5

6 Geometry Regents Exam The coordinates of the vertices of RST are R(, 3), S(4, 4), and T(, ). 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.] 34 In the diagram below of circle O, diameter AB is perpendicular to chord CD at E. If AO = 10 and BE = 4, find the length of CE. 35 Triangle ABC has coordinates A(, ), B(, 1), and C(4, ). Triangle A B C is the image of ABC under T 5,. On the set of axes below, graph and label ABC and its image, A B C. Determine the relationship between the area of ABC and the area of A B C. Justify your response. 33 On the diagram below, use a compass and straightedge to construct the bisector of XYZ. [Leave all construction marks.] 6

7 Geometry Regents Exam A 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 1 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. 37 Triangle HKL has vertices H( 7, ), 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.] 38 In the diagram below of quadrilateral ABCD, AD BC and DAE BCE. Line segments AC, DB, and FG intersect at E. Prove: AEF CEG 7

8 ID: A 011ge Answer Section 1 ANS: 6x + 4 = 18x 1 54 = 1x x = 54 1 = 4.5 PTS: REF: 01101ge STA: G.G.35 TOP: Parallel Lines and Transversals ANS: 3 PTS: REF: 0110ge STA: G.G.1 TOP: Centroid, Orthocenter, Incenter and Circumcenter 3 ANS: PTS: REF: 01103ge STA: G.G.73 TOP: Equations of Circles 4 ANS: The diagonals of a rhombus are perpendicular. 180 (90 + 1) = 78 PTS: REF: 01104ge STA: G.G.39 TOP: Special Parallelograms 5 ANS: 1 d = (4 1) + (7 11) = = 5 = 5 PTS: REF: 01105ge STA: G.G.67 TOP: Distance KEY: general 6 ANS: PTS: REF: 01106ge STA: G.G.3 TOP: Exterior Angle Theorem 7 ANS: 1 PTS: REF: 01107ge STA: G.G.0 TOP: Constructions 8 ANS: 4 PTS: REF: 01108ge STA: G.G.53 TOP: Segments Intercepted by Circle KEY: two tangents 9 ANS: 3 PTS: REF: 01109ge STA: G.G.44 TOP: Similarity Proofs 10 ANS: = PTS: REF: 01110ge STA: G.G.30 TOP: Interior and Exterior Angles of Triangles 11 ANS: PTS: REF: 01111ge STA: G.G.55 TOP: Properties of Transformations 1 ANS: 4 PTS: REF: 0111ge STA: G.G.71 TOP: Equations of Circles 13 ANS: 1 PTS: REF: 01113ge STA: G.G.4 TOP: Negations 1

9 ID: A 14 ANS: 50 + x = x = 68 x = 18 PTS: REF: 01114ge STA: G.G.51 TOP: Arcs Determined by Angles KEY: inside circle 15 ANS: PTS: REF: 01115ge STA: G.G.1 TOP: Volume 16 ANS: 4 PTS: REF: 01116ge STA: G.G.9 TOP: Triangle Congruency 17 ANS: 3 PTS: REF: 01117ge STA: G.G.64 TOP: Parallel and Perpendicular Lines 18 ANS: 1 PTS: REF: 01118ge STA: G.G.3 TOP: Planes 19 ANS: = 4 PTS: REF: 01119ge STA: G.G.40 TOP: Trapezoids 0 ANS: 1 PTS: REF: 0110ge STA: G.G.7 TOP: Equations of Circles 1 ANS: 1 PTS: REF: 0111ge STA: G.G.10 TOP: Solids ANS: 4 PTS: REF: 011ge STA: G.G.34 TOP: Angle Side Relationship 3 ANS: 3 PTS: REF: 0113ge STA: G.G.36 TOP: Interior and Exterior Angles of Polygons 4 ANS: 3 y = mx + b 1 = () + b 5 = b PTS: REF: 0114ge STA: G.G.65 TOP: Parallel and Perpendicular Lines 5 ANS: 4 AB is a vertical line, so its perpendicular bisector is a horizontal line through the midpoint of AB, which is (0, 3). PTS: REF: 0115ge STA: G.G.68 TOP: Perpendicular Bisector

10 ID: A 6 ANS: 1 7x + 4 = (x + 5). PM = () + 5 = 9 7x + 4 = 4x x = 6 x = PTS: REF: 0116ge STA: G.G.43 TOP: Centroid 7 ANS: 4 x 4x = 6. PQ = 4x + x = 5x = 5(3) = 15 4x = 36 x = 3 PTS: REF: 0117ge STA: G.G.47 TOP: Similarity KEY: leg 8 ANS: 5 3 =, = 8 PTS: REF: 0118ge STA: G.G.33 TOP: Triangle Inequality Theorem 9 ANS: x 0 = x + 0. mab = x + 0 = = 60 x = 40 PTS: REF: 0119ge STA: G.G.5 TOP: Chords 30 ANS: PTS: REF: 01130ge STA: G.G. TOP: Locus 31 ANS: The slope of y = x + 3 is. The slope of y + x = 6 is A B = 1. Since the slopes are opposite reciprocals, the lines are perpendicular. PTS: REF: 01131ge STA: G.G.63 TOP: Parallel and Perpendicular Lines 3 ANS: R ( 3, ), S'( 4, 4), and T (, ). PTS: REF: 0113ge STA: G.G.54 TOP: Rotations 3

11 ID: A 33 ANS: PTS: REF: 01133ge STA: G.G.17 TOP: Constructions 34 ANS: EO = 6. CE = 10 6 = 8 PTS: REF: 01134ge STA: G.G.49 TOP: Chords 35 ANS: A (7, 4), B (7, 1). C (9, 4). The areas are equal because translations preserve distance. PTS: 4 REF: 01135ge STA: G.G.55 TOP: Properties of Transformations 36 ANS: V = πr h 600π = πr 1 50 = r 5 = r 5 = r. L = πrh = π PTS: 4 REF: 01136ge STA: G.G.14 TOP: Volume and Lateral Area 37 ANS: M 7 + 5, + 4 = M( 1, 3) N, = N(4, 0). MN is a midsegment. PTS: 4 REF: 01137ge STA: G.G.4 TOP: Midsegments 4

12 ID: A 38 ANS: Quadrilateral ABCD, AD BC and DAE BCE are given. AD BC because if two lines are cut by a transversal so that a pair of alternate interior angles are congruent, the lines are parallel. ABCD is a parallelogram because if one pair of opposite sides of a quadrilateral are both congruent and parallel, the quadrilateral is a parallelogram. AE CE because the diagonals of a parallelogram bisect each other. FEA GEC as vertical angles. AEF CEG by ASA. PTS: 6 REF: 01138ge STA: G.G.7 TOP: Quadrilateral Proofs 5

### 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.

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. Which value of x would prove l m? 1) 2.5 2) 4.5 3)

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