# 0616geo. Geometry CCSS Regents Exam x 2 + 4x = (y 2 20)

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1 0616geo 1 A student has a rectangular postcard that he folds in half lengthwise. Next, he rotates it continuously about the folded edge. Which three-dimensional object below is generated by this rotation? A three-inch line segment is dilated by a scale factor of 6 and centered at its midpoint. What is the length of its image? 1) 9 inches ) inches ) 15 inches 4) 18 inches 1) ) Kevin s work for deriving the equation of a circle is shown below. x + 4x = (y 0) STEP 1 x + 4x = y + 0 STEP x + 4x + 4 = y STEP (x + ) = y STEP 4 (x + ) + y = 16 In which step did he make an error in his work? 1) Step 1 ) Step ) Step 4) Step 4 ) 4) 1

2 4 Which transformation of OA would result in an image parallel to OA? 5 Using the information given below, which set of triangles can not be proven similar? 1) 1) a translation of two units down ) a reflection over the x-axis ) a reflection over the y-axis 4) a clockwise rotation of 90 about the origin ) ) 4)

3 6 A company is creating an object from a wooden cube with an edge length of 8.5 cm. A right circular cone with a diameter of 8 cm and an altitude of 8 cm will be cut out of the cube. Which expression represents the volume of the remaining wood? 1) (8.5) π(8) (8) ) (8.5) π(4) (8) 9 In parallelogram ABCD, diagonals AC and BD intersect at E. Which statement does not prove parallelogram ABCD is a rhombus? 1) AC DB ) AB BC ) AC DB 4) AC bisects DCB ) (8.5) 1 π(8) (8) 4) (8.5) 1 π(4) (8) 10 In the diagram below of circle O, OB and OC are radii, and chords AB, BC, and AC are drawn. 7 Two right triangles must be congruent if 1) an acute angle in each triangle is congruent ) the lengths of the hypotenuses are equal ) the corresponding legs are congruent 4) the areas are equal 8 Which sequence of transformations will map ABC onto A' B' C'? Which statement must always be true? 1) BAC BOC ) m BAC = 1 m BOC ) BAC and BOC are isosceles. 4) The area of BAC is twice the area of BOC. 11 A 0-foot support post leans against a wall, making a 70 angle with the ground. To the nearest tenth of a foot, how far up the wall will the support post reach? 1) 6.8 ) 6.9 ) ) ) reflection and translation ) rotation and reflection ) translation and dilation 4) dilation and rotation

4 1 Line segment NY has endpoints N( 11,5) and Y(5, 7). What is the equation of the perpendicular bisector of NY? 1) y + 1 = 4 (x + ) 14 In the diagram below, ABC has vertices A(4,5), B(,1), and C(7,). ) y + 1 = (x + ) 4 ) y 6 = 4 (x 8) 4) y 6 = (x 8) 4 1 In RST shown below, altitude SU is drawn to RT at U. If SU = h, UT = 1, and RT = 4, which value of h will make RST a right triangle with RST as a right angle? 1) 6 ) 6 10 ) ) 6 5 What is the slope of the altitude drawn from A to BC? 1) 5 ) ) 1 4) 5 4

5 15 In the diagram below, ERM JTM. 17 In the diagram below, DB and AF intersect at point C, and AD and FBE are drawn. Which statement is always true? 1) cos J = RM RE ) cos R = JM JT ) tant = RM EM 4) tane = TM JM 16 On the set of axes below, rectangle ABCD can be proven congruent to rectangle KLMN using which transformation? If AC = 6, DC = 4, FC = 15, m D = 65, and m CBE = 115, what is the length of CB? 1) 10 ) 1 ) 17 4).5 18 Seawater contains approximately 1. ounces of salt per liter on average. How many gallons of seawater, to the nearest tenth of a gallon, would contain 1 pound of salt? 1). ).5 ) 4.7 4) 1. 1) rotation ) translation ) reflection over the x-axis 4) reflection over the y-axis 5

6 19 Line segment EA is the perpendicular bisector of ZT, and ZE and TE are drawn. 1 In the diagram of ABC, points D and E are on AB and CB, respectively, such that AC DE. Which conclusion can not be proven? 1) EA bisects angle ZET. ) Triangle EZT is equilateral. ) EA is a median of triangle EZT. 4) Angle Z is congruent to angle T. If AD = 4, DB = 1, and DE = 4, what is the length of AC? 1) 8 ) 1 ) 16 4) 7 Triangle RST is graphed on the set of axes below. 0 A hemispherical water tank has an inside diameter of 10 feet. If water has a density of 6.4 pounds per cubic foot, what is the weight of the water in a full tank, to the nearest pound? 1) 16,6 ),67 ) 10,690 4) 61,81 How many square units are in the area of 1) ) ) 45 4) 90 RST? 6

7 The graph below shows AB, which is a chord of circle O. The coordinates of the endpoints of AB are A(,) and B(, 7). The distance from the midpoint of AB to the center of circle O is units. 5 Describe a sequence of transformations that will map ABC onto DEF as shown below. What could be a correct equation for circle O? 1) (x 1) + (y + ) = 9 ) (x + 5) + (y ) = 9 ) (x 1) + (y ) = 5 4) (x 5) + (y + ) = 5 6 Point P is on segment AB such that AP:PB is 4:5. If A has coordinates (4,), and B has coordinates (,), determine and state the coordinates of P. 4 What is the area of a sector of a circle with a radius of 8 inches and formed by a central angle that measures 60? 8π 1) 16π ) π ) 64π 4) 7 In CED as shown below, points A and B are located on sides CE and ED, respectively. Line segment AB is drawn such that AE =.75, AC = 5, EB = 4.5, and BD = 6. Explain why AB is parallel to CD. 7

8 8 Find the value of R that will make the equation sin 7 =cos R true when 0 <R < 90. Explain your answer. 1 In the diagram below, radius OA is drawn in circle O. Using a compass and a straightedge, construct a line tangent to circle O at point A. [Leave all construction marks.] 9 In the diagram below, Circle 1 has radius 4, while Circle has radius 6.5. Angle A intercepts an arc of length π, and angle B intercepts an arc of length 1π 8. Dominic thinks that angles A and B have the same radian measure. State whether Dominic is correct or not. Explain why. 0 A ladder leans against a building. The top of the ladder touches the building 10 feet above the ground. The foot of the ladder is 4 feet from the building. Find, to the nearest degree, the angle that the ladder makes with the level ground. A barrel of fuel oil is a right circular cylinder where the inside measurements of the barrel are a diameter of.5 inches and a height of.5 inches. There are 1 cubic inches in a liquid gallon. Determine and state, to the nearest tenth, the gallons of fuel that are in a barrel of fuel oil. 8

9 Given: Parallelogram ABCD, EFG, and diagonal DFB 5 Given: Quadrilateral ABCD with diagonals AC and BD that bisect each other, and 1 Prove: ACD is an isosceles triangle and AEB is a right triangle Prove: DEF BGF 4 In the diagram below, A' B' C' is the image of ABC after a transformation. 6 A water glass can be modeled by a truncated right cone (a cone which is cut parallel to its base) as shown below. Describe the transformation that was performed. Explain why A' B' C' ABC. The diameter of the top of the glass is inches, the diameter at the bottom of the glass is inches, and the height of the glass is 5 inches. The base with a diameter of inches must be parallel to the base with a diameter of inches in order to find the height of the cone. Explain why. Determine and state, in inches, the height of the larger cone. Determine and state, to the nearest tenth of a cubic inch, the volume of the water glass. 9

10 ID: A 0616geo Answer Section 1 ANS: PTS: REF: geo NAT: G.GMD.B.4 TOP: Rotations of Two-Dimensional Objects ANS: 4 6 = 18 PTS: REF: 06160geo NAT: G.SRT.A.1 TOP: Line Dilations ANS: PTS: REF: 06160geo NAT: G.GPE.A.1 TOP: Equations of Circles KEY: find center and radius completing the square 4 ANS: 1 PTS: REF: geo NAT: G.CO.A. TOP: Identifying Transformations KEY: graphics 5 ANS: 1) 1 9 = 4 ) AA ) ) SAS PTS: REF: geo NAT: G.SRT.B.5 TOP: Similarity KEY: basic 6 ANS: 4 PTS: REF: geo NAT: G.GMD.A. TOP: Volume KEY: compositions 7 ANS: 1) only proves AA; ) need congruent legs for HL; ) SAS; 4) only proves product of altitude and base is equal PTS: REF: geo NAT: G.SRT.B.5 TOP: Triangle Proofs KEY: statements 8 ANS: 4 PTS: REF: geo NAT: G.SRT.A. TOP: Compositions of Transformations KEY: grids 9 ANS: 1 1) opposite sides; ) adjacent sides; ) perpendicular diagonals; 4) diagonal bisects angle PTS: REF: geo NAT: G.CO.C.11 TOP: Special Quadrilaterals 10 ANS: PTS: REF: geo NAT: G.C.A. TOP: Chords, Secants and Tangents KEY: inscribed 11 ANS: 4 sin 70 = x 0 x 18.8 PTS: REF: geo NAT: G.SRT.C.8 TOP: Using Trigonometry to Find a Side KEY: without graphics 1 ANS: 1 m = , = (, 1) m = = 1 16 = 4 m = 4 PTS: REF: 06161geo NAT: G.GPE.B.5 TOP: Parallel and Perpendicular Lines KEY: perpendicular bisector 1

11 ID: A 1 ANS: h = 0 1 h = 60 h = 6 10 PTS: REF: 06161geo NAT: G.SRT.B.5 TOP: Similarity KEY: altitude 14 ANS: 4 The slope of BC is 5. Altitude is perpendicular, so its slope is 5. PTS: REF: geo NAT: G.GPE.B.5 TOP: Parallel and Perpendicular Lines KEY: find slope of perpendicular line 15 ANS: 4 PTS: REF: geo NAT: G.SRT.C.6 TOP: Trigonometric Ratios 16 ANS: PTS: REF: geo NAT: G.CO.A. TOP: Identifying Transformations KEY: graphics 17 ANS: 1 f 4 = 15 6 f = 10 PTS: REF: geo NAT: G.CO.C.9 TOP: Lines and Angles 18 ANS: 1l 16 oz 1. oz 1 lb = 1. l 1. l 1g lb lb.785 l.5 g 1 lb PTS: REF: geo NAT: G.MG.A. TOP: Density 19 ANS: PTS: REF: geo NAT: G.CO.C.10 TOP: Triangle Proofs 0 ANS: π ,6 PTS: REF: 06160geo NAT: G.MG.A. TOP: Density

12 ID: A 1 ANS: 1 4 = 6 x 1x = 144 x = 1 PTS: REF: 06161geo NAT: G.SRT.B.5 TOP: Side Splitter Theorem ANS: 45 = = 6 5 a = = 1 (18)(5) = 45 PTS: REF: 0616geo NAT: G.GPE.B.7 TOP: Polygons in the Coordinate Plane ANS: 1 r = + 5 = 9 Since the midpoint of AB is (, ), the center must be either (5, ) or (1, ). PTS: REF: 0616geo NAT: G.GPE.A.1 TOP: Equations of Circles KEY: other 4 ANS: π = 1 π 64π = 6 PTS: REF: 06164geo NAT: G.C.B.5 TOP: Sectors 5 ANS: T 6,0 r x-axis PTS: REF: 06165geo NAT: G.CO.A.5 TOP: Compositions of Transformations KEY: identify

13 ID: A 6 ANS: ( 4) ( ) 9 (1,) (18) (0) + 0 PTS: REF: 06166geo NAT: G.GPE.B.6 TOP: Directed Line Segments 7 ANS:.75 5 = AB is parallel to CD because AB divides the sides proportionately = 9.75 PTS: REF: 06167geo NAT: G.SRT.B.5 TOP: Side Splitter Theorem 8 ANS: 7 + R = 90 Equal cofunctions are complementary. R = 17 PTS: REF: 06168geo NAT: G.SRT.C.7 TOP: Cofunctions 9 ANS: s = θ r s = θ r Yes, both angles are equal. π = A 4 π 4 = A 1π 8 = B 6.5 π 4 = B PTS: REF: 06169geo NAT: G.C.B.5 TOP: Arc Length KEY: arc length 0 ANS: tanx = 10 4 x 68 PTS: REF: 06160geo NAT: G.SRT.C.8 TOP: Using Trigonometry to Find an Angle 4

14 ID: A 1 ANS: PTS: REF: 06161geo NAT: G.CO.D.1 TOP: Constructions KEY: parallel and perpendicular lines ANS: π PTS: 4 REF: 0616geo NAT: G.GMD.A. TOP: Volume KEY: cylinders ANS: Parallelogram ABCD, EFG, and diagonal DFB (given); DFE BFG (vertical angles); AD CB (opposite sides of a parallelogram are parallel); EDF GBF (alternate interior angles are congruent); DEF BGF (AA). PTS: 4 REF: 0616geo NAT: G.SRT.A. TOP: Similarity Proofs 4 ANS: A dilation of 5 about the origin. Dilations preserve angle measure, so the triangles are similar by AA. PTS: 4 REF: 06164geo NAT: G.SRT.A. TOP: Similarity Proofs 5 ANS: Quadrilateral ABCD with diagonals AC and BD that bisect each other, and 1 (given); quadrilateral ABCD is a parallelogram (the diagonals of a parallelogram bisect each other); AB CD (opposite sides of a parallelogram are parallel); 1 and 4 (alternate interior angles are congruent); and 4 (substitution); ACD is an isosceles triangle (the base angles of an isosceles triangle are congruent); AD DC (the sides of an isosceles triangle are congruent); quadrilateral ABCD is a rhombus (a rhombus has consecutive congruent sides); AE BE (the diagonals of a rhombus are perpendicular); BEA is a right angle (perpendicular lines form a right angle); AEB is a right triangle (a right triangle has a right angle). PTS: 6 REF: 06165geo NAT: G.CO.C.11 TOP: Quadrilateral Proofs 5

15 ID: A 6 ANS: Similar triangles are required to model and solve a proportion. x = x 1 x + 5 = 1.5x 5 =.5x 10 = x = 15 1 π(1.5) (15) 1 π(1) (10) 4.9 PTS: 6 REF: 06166geo NAT: G.GMD.A. TOP: Volume KEY: cones 6

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