# 0110ge. Geometry Regents Exam Which expression best describes the transformation shown in the diagram below?

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1 0110ge 1 In the diagram below of trapezoid RSUT, RS TU, X is the midpoint of RT, and V is the midpoint of SU. 3 Which expression best describes the transformation shown in the diagram below? If RS = 30 and XV = 44, what is the length of TU? 1) 37 ) 58 3) 74 4) 118 In ABC, m A = x, m B = x +, and m C = 3x + 4. What is the value of x? 1) 9 ) 31 3) 59 4) 61 1) same orientation; reflection ) opposite orientation; reflection 3) same orientation; translation 4) opposite orientation; translation 1

2 4 Based on the construction below, which statement must be true? 6 Which transformation is not always an isometry? 1) rotation ) dilation 3) reflection 4) translation 7 In ABC, AB BC. An altitude is drawn from B to AC and intersects AC at D. Which conclusion is not always true? 1) ABD CBD ) BDA BDC 3) AD BD 4) AD DC 1) m ABD = 1 m CBD ) m ABD = m CBD 3) m ABD = m ABC 8 In the diagram below, tangent PA and secant PBC are drawn to circle O from external point P. 4) m CBD = 1 m ABD 5 In the diagram below, ABC is inscribed in circle P. The distances from the center of circle P to each side of the triangle are shown. If PB = 4 and BC = 5, what is the length of PA? 1) 0 ) 9 3) 8 4) 6 Which statement about the sides of the triangle is true? 1) AB > AC > BC ) AB < AC and AC > BC 3) AC > AB > BC 4) AC = AB and AB > BC

3 9 Which geometric principle is used to justify the construction below? 1 Lines j and k intersect at point P. Line m is drawn so that it is perpendicular to lines j and k at point P. Which statement is correct? 1) Lines j and k are in perpendicular planes. ) Line m is in the same plane as lines j and k. 3) Line m is parallel to the plane containing lines j and k. 4) Line m is perpendicular to the plane containing lines j and k. 1) A line perpendicular to one of two parallel lines is perpendicular to the other. ) Two lines are perpendicular if they intersect to form congruent adjacent angles. 3) When two lines are intersected by a transversal and alternate interior angles are congruent, the lines are parallel. 4) When two lines are intersected by a transversal and the corresponding angles are congruent, the lines are parallel. 13 In the diagram below of parallelogram STUV, SV = x + 3, VU = x 1, and TU = 4x Which equation represents the circle whose center is (, 3) and whose radius is 5? 1) (x ) + (y + 3) = 5 ) (x + ) + (y 3) = 5 3) (x + ) + (y 3) = 5 4) (x ) + (y + 3) = 5 11 Towns A and B are 16 miles apart. How many points are 10 miles from town A and 1 miles from town B? 1) 1 ) 3) 3 4) 0 What is the length of SV? 1) 5 ) 3) 7 4) 4 14 Which equation represents a line parallel to the line whose equation is y 5x = 10? 1) 5y x = 5 ) 5y + x = 10 3) 4y 10x = 1 4) y + 10x = 8 3

4 15 In the diagram below of circle O, chords AD and BC intersect at E, mac = 87, and mbd = What is an equation of the line that contains the point (3, 1) and is perpendicular to the line whose equation is y = 3x +? 1) y = 3x + 8 ) y = 3x 3) y = 1 3 x 4) y = 1 3 x What is the degree measure of CEA? 1) 87 ) 61 3) ) 6 19 In the diagram below, SQ and PR intersect at T, PQ is drawn, and PS QR. 16 In the diagram below of ADB, m BDA = 90, AD = 5, and AB = 15. What is the length of BD? 1) 10 ) 0 3) 50 4) 110 What technique can be used to prove that PST RQT? 1) SAS ) SSS 3) ASA 4) AA 17 What is the distance between the points ( 3, ) and (1, 0)? 1) ) 3 3) 5 4) 5 4

5 0 The equation of a circle is (x ) + (y + 4) = 4. Which diagram is the graph of the circle? 1 In the diagram below, ABC is shown with AC extended through point D. 1) If m BCD = 6x +, m BAC = 3x + 15, and m ABC = x 1, what is the value of x? 1) 1 ) ) 16 4) ) 3) Given ABC DEF such that AB DE = 3. Which statement is not true? BC 1) EF = 3 m A ) m D = 3 3) area of ABC area of DEF = 9 4 4) perimeter of ABC perimeter of DEF = 3 4) 5

6 3 The pentagon in the diagram below is formed by five rays. 6 What is the image of point A(4, ) after the composition of transformations defined by R 90 r y = x? 1) ( 4, ) ) (4, ) 3) ( 4, ) 4) (, 4) 7 Which expression represents the volume, in cubic centimeters, of the cylinder represented in the diagram below? What is the degree measure of angle x? 1) 7 ) 96 3) 108 4) 11 4 Through a given point, P, on a plane, how many lines can be drawn that are perpendicular to that plane? 1) 1 ) 3) more than 4) none 5 What is the slope of a line that is perpendicular to the line whose equation is 3x + 4y = 1? 3 1) 4 ) ) 3 4) 4 3 1) 16π ) 34π 3) 97π 4) 3, 888π 8 What is the inverse of the statement If two triangles are not similar, their corresponding angles are not congruent? 1) If two triangles are similar, their corresponding angles are not congruent. ) If corresponding angles of two triangles are not congruent, the triangles are not similar. 3) If two triangles are similar, their corresponding angles are congruent. 4) If corresponding angles of two triangles are congruent, the triangles are similar. 6

7 9 In RST, m RST = 46 and RS ST. Find m STR. 30 Tim has a rectangular prism with a length of 10 centimeters, a width of centimeters, and an unknown height. He needs to build another rectangular prism with a length of 5 centimeters and the same height as the original prism. The volume of the two prisms will be the same. Find the width, in centimeters, of the new prism. 33 In the diagram below of ACD, E is a point on AD and B is a point on AC, such that EB DC. If AE = 3, ED = 6, and DC = 15, find the length of EB. 31 In the diagram below of circle C, QR is a diameter, and Q(1, 8) and C(3.5, ) are points on a coordinate plane. Find and state the coordinates of point R. 34 In the diagram below of TEM, medians TB, EC, and MA intersect at D, and TB = 9. Find the length of TD. 3 Using a compass and straightedge, and AB below, construct an equilateral triangle with all sides congruent to AB. [Leave all construction marks.] 35 In KLM, m K = 36 and KM = 5. The transformation D is performed on KLM to form K L M. Find m K. Justify your answer. Find the length of K M. Justify your answer. 7

8 36 Given: JKLM is a parallelogram. JM LN LMN LNM Prove: JKLM is a rhombus. 38 On the set of axes below, solve the following system of equations graphically for all values of x and y. y = (x ) + 4 4x + y = On the grid below, graph the points that are equidistant from both the x and y axes and the points that are 5 units from the origin. Label with an X all points that satisfy both conditions. 8

9 ID: A 0110ge Answer Section 1 ANS: The length of the midsegment of a trapezoid is the average of the lengths of its bases. x + 30 = 44. x + 30 = 88 x = 58 PTS: REF: ge STA: G.G.40 TOP: Trapezoids ANS: 1 x + x + + 3x + 4 = 180 6x + 6 = 180 x = 9 PTS: REF: 01100ge STA: G.G.30 TOP: Interior and Exterior Angles of Triangles 3 ANS: PTS: REF: ge STA: G.G.55 TOP: Properties of Transformations 4 ANS: PTS: REF: ge STA: G.G.17 TOP: Constructions 5 ANS: 1 The closer a chord is to the center of a circle, the longer the chord. PTS: REF: ge STA: G.G.49 TOP: Chords 6 ANS: PTS: REF: ge STA: G.G.56 TOP: Isometries 7 ANS: 3 PTS: REF: ge STA: G.G.31 TOP: Isosceles Triangle Theorem 8 ANS: 4 x = (4 + 5) 4 x = 36 x = 6 PTS: REF: ge STA: G.G.53 TOP: Segments Intercepted by Circle KEY: tangent and secant 9 ANS: 4 PTS: REF: ge STA: G.G.19 TOP: Constructions 10 ANS: 3 PTS: REF: ge STA: G.G.71 TOP: Equations of Circles 11 ANS: PTS: REF: ge STA: G.G. TOP: Locus 1 ANS: 4 PTS: REF: 01101ge STA: G.G.1 TOP: Planes 1

10 ID: A 13 ANS: 1 Opposite sides of a parallelogram are congruent. 4x 3 = x + 3. SV = () + 3 = 5. 3x = 6 x = PTS: REF: ge STA: G.G.38 TOP: Parallelograms 14 ANS: 3 m = A B = 5. m = A B = 10 4 = 5 PTS: REF: ge STA: G.G.63 TOP: Parallel and Perpendicular Lines 15 ANS: = 1 = 61 PTS: REF: ge STA: G.G.51 TOP: Arcs Determined by Angles KEY: inside circle 16 ANS: 1 a + (5 ) = ( 15 ) a + (5 ) = 4 15 a + 50 = 60 a = 10 a = 10 PTS: REF: ge STA: G.G.48 TOP: Pythagorean Theorem 17 ANS: 4 d = ( 3 1) + ( 0) = = 0 = 4 5 = 5 PTS: REF: ge STA: G.G.67 TOP: Distance 18 ANS: 4 The slope of y = 3x + is 3. The perpendicular slope is = 1 3 (3) + b 1 = 1 + b b = PTS: REF: ge STA: G.G.64 TOP: Parallel and Perpendicular Lines 19 ANS: 4 PTS: REF: ge STA: G.G.44 TOP: Similarity Proofs 0 ANS: PTS: REF: 01100ge STA: G.G.74 TOP: Graphing Circles

11 ID: A 1 ANS: 1 3x x 1 = 6x + 5x + 14 = 6x + x = 1 PTS: REF: 01101ge STA: G.G.3 TOP: Exterior Angle Theorem ANS: Because the triangles are similar, m A m D = 1 PTS: REF: 0110ge STA: G.G.45 TOP: Similarity KEY: perimeter and area 3 ANS: 3. The sum of the interior angles of a pentagon is (5 )180 = 540. PTS: REF: 01103ge STA: G.G.36 TOP: Interior and Exterior Angles of Polygons 4 ANS: 1 PTS: REF: 01104ge STA: G.G.3 TOP: Planes 5 ANS: 3 m = A B = 3 4 PTS: REF: 01105ge STA: G.G.6 TOP: Parallel and Perpendicular Lines 6 ANS: 1 A (, 4) PTS: REF: 01103ge STA: G.G.54 TOP: Compositions of Transformations KEY: basic 7 ANS: 3 V = πr h = π 6 7 = 97π PTS: REF: 01107ge STA: G.G.14 TOP: Volume and Lateral Area 8 ANS: 3 PTS: REF: 01108ge STA: G.G.6 TOP: Conditional Statements 3

12 ID: A 9 ANS: = 67 PTS: REF: 01109ge STA: G.G.31 TOP: Isosceles Triangle Theorem 30 ANS: 4. l 1 w 1 h 1 = l w h 10 h = 5 w h 0 = 5w w = 4 PTS: REF: ge STA: G.G.11 TOP: Volume 31 ANS: (6, 4). C x = Q x + R x. C y = Q y + R y. 3.5 = 1 + R x = 8 + R y 7 = 1 + R x 6 = R x 4 = 8 + R y 4 = R y PTS: REF: ge STA: G.G.66 TOP: Midpoint 3 ANS: PTS: REF: 01103ge STA: G.G.0 TOP: Constructions 33 ANS: 5. 3 x = x = 45 x = 5 PTS: REF: ge STA: G.G.46 TOP: Side Splitter Theorem 34 ANS: 6. The centroid divides each median into segments whose lengths are in the ratio : 1. TD = 6 and DB = 3 PTS: REF: ge STA: G.G.43 TOP: Centroid 4

13 ID: A 35 ANS: 36, because a dilation does not affect angle measure. 10, because a dilation does affect distance. PTS: 4 REF: ge STA: G.G.59 TOP: Properties of Transformations 36 ANS: JK LM because opposite sides of a parallelogram are congruent. LM LN because of the Isosceles Triangle Theorem. LM JM because of the transitive property. JKLM is a rhombus because all sides are congruent. PTS: 4 REF: ge STA: G.G.7 TOP: Quadrilateral Proofs 37 ANS: PTS: 4 REF: ge STA: G.G.3 TOP: Locus 38 ANS: PTS: 6 REF: ge STA: G.G.70 TOP: Quadratic-Linear Systems 5

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