# Geometry Regents Practice Midterm

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1 Class: Date: Geometry Regents Practice Midterm Multiple Choice Identify the choice that best completes the statement or answers the question. 1. ( points) What is the equation of a line that is parallel to the line whose equation is y = x +? a. x + y = 5 b. x + y = c. y x = 1 d. y x = 3. ( points) The diagram below shows the construction of a line through point P perpendicular to line m. Which statement is demonstrated by this construction? a. If a line is parallel to a line that is perpendicular to a third line, then the line is also perpendicular to the third line. b. The set of points equidistant from the endpoints of a line segment is the perpendicular bisector of the segment. c. Two lines are perpendicular if they are equidistant from a given point. d. Two lines are perpendicular if they intersect to form a vertical line. 3. ( points) A transversal intersects two lines. Which condition would always make the two lines parallel? a. Vertical angles are congruent. b. Alternate interior angles are congruent. c. Corresponding angles are supplementary. d. Same-side interior angles are complementary. 1

2 4. ( points) In the diagram below of ABC, side BC is extended to point D, m A = x, m B = x + 15, and m ACD = 5x + 5. What is m B? a. 5 c. 5 b. 0 d (1 point) M is the midpoint of AN, A has coordinates ( 6, 6), and M has coordinates (1, ). Find the coordinates of N. a. (8, 10) b. ( 5, 4) c. ( 1, ) d. (8 1, 91 ) 6. ( points) Given the equations: y = x 6x + 10 y + x = 4 What is the solution to the given system of equations? a. (,3) b. (3,) c. (,) and (1,3) d. (,) and (3,1) 7. ( points) What is an equation of the line that passes through the point (7,3) and is parallel to the line 4x + y = 10? a. y = 1 x 1 b. y = 1 x + 13 c. y = x 11 d. y = x + 17

3 8. ( points) In plane P, lines m and n intersect at point A. If line k is perpendicular to line m and line n at point A, then line k is a. contained in plane P b. parallel to plane P c. perpendicular to plane P d. skew to plane P 9. ( points) The lines represented by the equations y + 1 x = 4 and 3x + 6y = 1 are a. the same line b. parallel c. perpendicular d. neither parallel nor perpendicular 10. ( points) What is the slope of a line perpendicular to the line whose equation is y = 3 x 5? a. 3 b. 3 c. 3 d ( points) In the diagram below, line p intersects line m and line n. If m 1 = 7x and m = 5x + 30, lines m and n are parallel when x equals a. 1.5 b. 15 c d

4 1. ( points) Which equation represents a line perpendicular to the line whose equation is y 5x = 10? a. 5y x = 5 b. 5y + x = 10 c. 4y 10x = 1 d. y + 10x = ( points) In the diagram of quadrilateral ABCD, AB Ä CD, ABC CDA, and diagonal AC is drawn. Which method can be used to prove ABC is congruent to CDA? a. AAS b. SSA c. SAS d. SSS 14. ( points) A support beam between the floor and ceiling of a house forms a 90º angle with the floor. The builder wants to make sure that the floor and ceiling are parallel. Which angle should the support beam form with the ceiling? a. 45º b. 60º c. 90º d. 180º 15. ( points) If two different lines are perpendicular to the same plane, they are a. collinear b. coplanar c. congruent d. consecutive 4

5 16. ( points) What is the value of x, in inches, in the right triangle below? a. 15 b. 8 c. 34 d ( points) In the diagram of ABC below, AB AC. The measure of B is 40. What is the measure of A? a. 40 b. 50 c. 70 d

6 18. ( points) In the diagram below of ABC, D is the midpoint of AB, and E is the midpoint of BC. If AC = 4x + 10, which expression represents DE? a. x +.5 b. x + 5 c. x + 10 d. 8x ( points) In the diagram below of drawn. ABC, CD is the bisector of BCA, AE is the bisector of CAB, and BG is Which statement must be true? a. DG = EG b. AG = BG c. AEB AEC d. DBG EBG 6

7 0. ( points) In PQR, PQ = 8, QR = 1, and RP = 13. Which statement about the angles of PQR must be true? a. m Q > m P > m R b. m Q > m R > m P c. m R > m P > m Q d. m P > m R > m Q 1. ( points) The endpoints of CD are C(, 4) and D(6,). What are the coordinates of the midpoint of CD? a. (,3) b. (, 1) c. (4, ) d. (4,3). ( points) In the diagram below of AGE and OLD, GAE LOD, and AE OD. To prove that AGE and OLD are congruent by SAS, what other information is needed? a. GE LD b. AG OL c. AGE OLD d. AEG ODL 3. ( points) What is the converse of the statement "If Bob does his homework, then George gets candy"? a. If George gets candy, then Bob does his homework. b. Bob does his homework if and only if George gets candy. c. If George does not get candy, then Bob does not do his homework. d. If Bob does not do his homework, then George does not get candy. 4. (1 point) For a triangle, list the respective names of the points of concurrency of perpendicular bisectors of the sides bisectors of the angles medians lines containing the altitudes. a. incenter circumcenter centroid orthocenter b. circumcenter incenter centroid orthocenter c. circumcenter incenter orthocenter centroid d. incenter circumcenter orthocenter centroid 7

8 5. ( points) In ABC, m A = x, m B = x +, and m C = 3x + 4. What is the value of x? a. 9 b. 31 c. 59 d ( points) The diagram below shows the construction of the bisector of ABC. Which statement is not true? a. m EBF = 1 m ABC c. m EBF = m ABC b. m DBF = 1 m ABC d. m DBF = m EBF 7. ( points) What is the length of the line segment with endpoints ( 6,4) and (, 5)? a. 13 b. 17 c. 7 d ( points) What is the contrapositive of the statement, If I am tall, then I will bump my head? a. If I bump my head, then I am tall. b. If I do not bump my head, then I am tall. c. If I am tall, then I will not bump my head. d. If I do not bump my head, then I am not tall. 8

9 Short Answer 9. ( points) On the diagram below, use a compass and straightedge to construct the bisector of ABC. [Leave all construction marks.] 30. ( points) In the diagram of KLM below, m L = 70, m M = 50, and MK is extended through N. What is the measure of LKN? 31. ( points) Find an equation of the line passing through the point (5,4) and parallel to the line whose equation is x + y = 3. Answer: 9

10 3. ( points) In right DEF, m D = 90 and m F is 1 degrees less than twice m E. Find m E. Answer: 33. ( points) DFG and JKL are complementary angles. m DFG = x + 5, and m JKL = x 9. Find the measure of each angle. Answer: 34. ( points) Circle the set of numbers that represents the lengths of the sides of a triangle. Justify your answer in the space provided. {5,18,13} {6,17,} {16,4,7} {6,8,15} 10

11 35. (4 points) Write an equation of the perpendicular bisector of the line segment whose endpoints are ( 1,1) and (7, 5). [The use of the grid below is optional] Answer: 36. (4 points) Solve the following system of equations graphically. y = x 4x 1 y = x + 1 Solution: 11

12 37. (4 points) In the diagram below of quadrilateral ABCD with diagonal BD, m A = 93, m ADB = 43, m C = 3x + 5, m BDC = x + 19, and m DBC = x + 6. Determine if AB is parallel to DC. Explain your reasoning. 38. (6 points) Given: ABC and EDC, C is the midpoint of BD and AE Prove: AB Ä DE 1

13 Geometry Regents Practice Midterm Answer Section MULTIPLE CHOICE 1. ANS: C The slope of y = x + is 1. The slope of y x = 1 is A B = ( 1) = 1. 1 PTS: REF: ge STA: G.G.63 TOP: Parallel and Perpendicular Lines. ANS: B PTS: REF: 06100ge STA: G.G.19 TOP: Constructions 3. ANS: B PTS: REF: ge STA: G.G.35 TOP: Parallel Lines and Transversals 4. ANS: C x + x + 15 = 5x + 15 (5) + 15 = 5 3x + 15 = 5x = x 5 = x PTS: REF: 01117ge STA: G.G.3 TOP: Exterior Angle Theorem 1

14 5. ANS: A Step 1 Let the coordinates of N equal (x, y). Step Use the Midpoint Formula. Ê Ê Ë Á 1, ˆ = x + x 1 y 1 + y ˆ Ê, Ë Á = Ë Á Step 3 Find the x- and y-coordinates. 6 + x, 6 + y 1 = 6 + x = 6 + y Set the coordinates equal. Ê ( 1) = 6 + x ˆ Ê ( ) = 6 + y ˆ Ë Á Ë Á Multiply both sides by. = 6 + x 4 = 6 + y Simplify. x = 8 y = 10 Solve for x or y, as appropriate. The coordinates of N are (8, 10). ˆ A B C D Feedback Correct! Let the coordinates of N be (x, y). Substitute known values into the Midpoint Formula to solve for x and y. This is the midpoint of line segment AM. If M is the midpoint of line segment AN, what are the coordinates of N? Let the coordinates of N be (x, y). Substitute known values into the Midpoint Formula to solve for x and y. PTS: 1 DIF: Average REF: 1954dafa df-9c7d f0dea OBJ: 1-6. Finding the Coordinates of an Endpoint STA: NY.NYLES.MTH.05.GEO.G.G.66 LOC: MTH.C TOP: 1-6 Midpoint and Distance in the Coordinate Plane KEY: coordinate geometry midpoint DOK: DOK 6. ANS: D y + x = 4. x 6x + 10 = x + 4. y + x = 4. y + = 4 y = x + 4 x 5x + 6 = 0 (x 3)(x ) = 0 x = 3 or y + 3 = 4 y = 1 y = PTS: REF: 08091ge STA: G.G.70 TOP: Quadratic-Linear Systems

15 7. ANS: D The slope of a line in standard form is A 4, so the slope of this line is =. A parallel line would also B have a slope of. Since the answers are in slope intercept form, find the y-intercept: y = mx + b 3 = (7) + b 17 = b PTS: REF: ge STA: G.G.65 TOP: Parallel and Perpendicular Lines 8. ANS: C PTS: REF: ge STA: G.G.1 TOP: Planes 9. ANS: B y + 1 x = 4 y = 1 x + 4 3x + 6y = 1 6y = 3x + 1 y = 3 6 x + m = 1 y = 1 x + PTS: REF: ge STA: G.G.63 TOP: Parallel and Perpendicular Lines 10. ANS: D The slope of y = 3 x 5 is. Perpendicular lines have slope that are opposite reciprocals. 3 PTS: REF: ge STA: G.G.6 TOP: Parallel and Perpendicular Lines 11. ANS: B 7x = 5x + 30 x = 30 x = 15 PTS: REF: ge STA: G.G.35 TOP: Parallel Lines and Transversals 1. ANS: B m = A B = 5. m = A B = 10 4 = 5 PTS: REF: ge STA: G.G.63 TOP: Parallel and Perpendicular Lines 13. ANS: A PTS: REF: 0111GE STA: G.G.8 TOP: Triangle Congruency 14. ANS: C PTS: REF: 08100ge STA: G.G.9 TOP: Planes 15. ANS: B PTS: REF: 08097ge STA: G.G.4 TOP: Planes 3

16 16. ANS: C = x 34 = x 34 = x PTS: REF: ia STA: A.A.45 TOP: Pythagorean Theorem 17. ANS: D 180 ( ) = 100 PTS: REF: ge STA: G.G.31 TOP: Isosceles Triangle Theorem 18. ANS: B 4x + 10 = x + 5 PTS: REF: ge STA: G.G.4 TOP: Midsegments 19. ANS: D BG is also an angle bisector since it intersects the concurrence of CD and AE PTS: REF: 06105ge STA: G.G.1 KEY: Centroid, Orthocenter, Incenter and Circumcenter 0. ANS: A PTS: REF: ge STA: G.G.34 TOP: Angle Side Relationship 1. ANS: B M x = + 6 =. M y = 4 + = 1 PTS: REF: ge STA: G.G.66 TOP: Midpoint KEY: general. ANS: B PTS: REF: ge STA: G.G.8 TOP: Triangle Congruency 3. ANS: A PTS: REF: ge STA: G.G.6 TOP: Converse and Biconditional 4. ANS: B PTS: 1 DIF: L3 REF: 5-3 Concurrent Lines, Medians, and Altitudes OBJ: 5-3. Medians and Altitudes NAT: NAEP 005 G3b STA: NY G.CM.6 NY G.G.1 NY G.G.43 KEY: angle bisector circumcenter of the triangle centroid orthocenter of the triangle median altitude perpendicular bisector 4

17 5. ANS: A x + x + + 3x + 4 = 180 6x + 6 = 180 x = 9 PTS: REF: 01100ge STA: G.G.30 TOP: Interior and Exterior Angles of Triangles 6. ANS: C PTS: REF: 08090ge STA: G.G.17 TOP: Constructions 7. ANS: D d = ( 6 ) + (4 ( 5)) = = 145 PTS: REF: ge STA: G.G.67 TOP: Distance KEY: general 8. ANS: D PTS: REF: ge STA: G.G.6 TOP: Conditional Statements SHORT ANSWER 9. ANS: PTS: REF: ge STA: G.G.17 TOP: Constructions 30. ANS: 10º PTS: REF: ge STA: G.G.3 TOP: Exterior Angle Theorem 31. ANS: y = x The slope of x + y = 3 is A B = =. y = mx + b 1. 4 = ( )(5) + b b = 14 PTS: REF: ge STA: G.G.65 TOP: Parallel and Perpendicular Lines 5

18 3. ANS: 34. x 1 + x + 90 = 180 3x + 78 = 90 3x = 10 x = 34 PTS: REF: ge STA: G.G.30 TOP: Interior and Exterior Angles of Triangles 33. ANS: DFG = 5, JKL = 38 PTS: DIF: L3 REF: 1-6 Measuring Angles OBJ: 1-6. Identifying Angle Pairs NAT: NAEP 005 M1e NAEP 005 M1f NAEP 005 G3g STA: NY G.CM.5 NY G.CN.3 KEY: complementary angles 34. ANS: {6,17,} {6,17,} because > PTS: REF: ge STA: G.G.33 TOP: Triangle Inequality Theorem 35. ANS: y = 4 3 x 6. M = x = 3 The perpendicular bisector goes through (3, ) and has a slope of 4 3. M y = 1 + ( 5) = m = 1 ( 5) 1 7 = 3 4 y y M = m(x x M ). y 1 = 4 (x ) 3 PTS: 4 REF: ge STA: G.G.68 TOP: Perpendicular Bisector 6

19 36. ANS: PTS: 4 REF: ge STA: G.G.70 TOP: Quadratic-Linear Systems 37. ANS: Yes, m ABD = m BDC = ( ) = 44 x x x + 5 = 180. Because alternate interior angles ABD and CDB are congruent, AB is parallel to DC. 6x + 30 = 180 6x = 150 x = 5 x + 19 = 44 PTS: 4 REF: ge STA: G.G.35 TOP: Parallel Lines and Transversals 38. ANS: AC EC and DC BC because of the definition of midpoint. ACB ECD because of vertical angles. ABC EDC because of SAS. CDE CBA because of CPCTC. BD is a transversal intersecting AB and ED. Therefore AB Ä DE because CDE and CBA are congruent alternate interior angles. PTS: 6 REF: ge STA: G.G.7 TOP: Triangle Proofs 7

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