Geometry Module 4 Unit 2 Practice Exam

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1 Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning and accurate labeling of an isoscles trapezoid in the coordinate plane? a. c. b. d. 1

2 Name: ID: A 2. Which diagram shows the most useful positioning of a rectangle in the first quadrant of a coordinate plane? a. c. b. d. Short Answer 3. Is TVS scalene, isosceles, or equilateral? The vertices are T(1,1), V(4,0), and S(2,4). 4. A quadrilateral has vertices ( 3, 1), (4, 5), ( 1, 5), and ( 3, 3). What special quadrilateral is formed by connecting the midpoints of the sides? 5. In the coordinate plane, three vertices of rectangle ABCD are A(0, 0), B(0, a), and D(b, 0). What are the coordinates of point C? 6. The vertices of the trapezoid are the origin along with A(4p, 4q), B(4r, 4q), and C(4s, 0). Find the midpoint of the midsegment of the trapezoid. 2

3 Name: ID: A 7. For the parallelogram, find coordinates for P without using any new variables. 8. For A( 1, 1), B(2, 1), and C(2, 1), find all locations of a fourth point, D, so that a parallelogram is formed using A, B, C, D in order as vertices. Plot each point D on a coordinate grid and draw the parallelogram. 9. The fact that the diagonals of a kite are perpendicular suggests a way to place a kite in the coordinate plane. Show this placement. Include labels for the kite vertices. 10. Show how to place a rhombus in the coordinate plane so that its diagonals lie along the axes. Label the vertices using as few variables as possible. 11. Find the lengths of the diagonals of this trapezoid. 12. In the coordinate plane, draw a square with sides 8n units long. Give coordinates for each vertex, and the coordinates of the point of intersection of the diagonals. 3

4 Name: ID: A Essay 13. Verify that parallelogram ABCD with vertices A( 5, 1), B( 9, 6), C( 1, 5), and D(3, 2) is a rhombus by showing that it is a parallelogram with perpendicular diagonals. 14. Find the midpoint of each side of the kite. Connect the midpoints. What is the most precise classification of the quadrilateral formed by connecting the midpoints of the sides of the kite? 15. Prove using coordinate geometry: The midpoints of the sides of a rhombus determine a rectangle. 16. Prove using coordinate geometry: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. 17. Write a coordinate proof of the following theorem: If a parallelogram is a rectangle, then its diagonals are congruent. 4

5 Name: ID: A Other 18. In the coordinate plane, draw JKL with J(2, 3), K(10, 4), and L(8, 9). Classify JKL. Explain. 19. In the coordinate plane, draw parallelogram ABCD with A( 5, 0), B(1, 7), C(8, 1), and D(2, 6).Then demonstrate that ABCD is a rectangle. 20. AC is a segment in the coordinate plane. Explain why sometimes it is a good idea to give points A and C the coordinates (2a, 2b) and (2c, 2d). 21. If you want to prove that the diagonals of a parallelogram bisect each other using coordinate geometry, how would you place the parallelogram on the coordinate plane? Give the coordinates of the vertices for the placement you choose. 22. Write the Given and Prove statements for a proof of the following theorem: If a quadrilateral is a square, then its diagonals are perpendicular. Square FGHK and its diagonals have been drawn for you. 23. Write a coordinate proof of the following theorem: If a quadrilateral is a kite, then its diagonals are perpendicular. 5

6 Geometry Module 4 Unit 2 Practice Exam Answer Section MULTIPLE CHOICE 1. ANS: A PTS: 1 DIF: L3 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 1 Naming Coordinates KEY: algebra coordinate plane isosceles trapezoid kite 2. ANS: A PTS: 1 DIF: L2 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 1 Naming Coordinates KEY: algebra coordinate plane rectangle square DOK: DOK 1 SHORT ANSWER 3. ANS: isosceles PTS: 1 DIF: L2 REF: 6-7 Polygons in the Coordinate Plane OBJ: Classify polygons in the coordinate plane STA: MA.912.G.1.1 MA.912.G.2.6 MA.912.G.3.1 MA.912.G.3.3 MA.912.G.4.1 MA.912.G.4.8 TOP: 6-7 Problem 1 Classifying a Triangle KEY: triangle distance formula isosceles scalene 4. ANS: kite PTS: 1 DIF: L3 REF: 6-7 Polygons in the Coordinate Plane OBJ: Classify polygons in the coordinate plane STA: MA.912.G.1.1 MA.912.G.2.6 MA.912.G.3.1 MA.912.G.3.3 MA.912.G.4.1 MA.912.G.4.8 TOP: 6-7 Problem 3 Classifying a Quadrilateral 5. ANS: (b, a) KEY: midpoint kite rectangle PTS: 1 DIF: L2 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 2 Using Variable Coordinates KEY: coordinate plane algebra rectangle 1

7 6. ANS: (p + r + s, 2q) PTS: 1 DIF: L3 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 2 Using Variable Coordinates KEY: algebra coordinate plane isosceles trapezoid midsegment 7. ANS: (a + c, b) PTS: 1 DIF: L2 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 2 Using Variable Coordinates KEY: parallelogram coordinate plane algebra 2

8 8. ANS: PTS: 1 DIF: L4 REF: 6-7 Polygons in the Coordinate Plane OBJ: Classify polygons in the coordinate plane STA: MA.912.G.1.1 MA.912.G.2.6 MA.912.G.3.1 MA.912.G.3.3 MA.912.G.4.1 MA.912.G.4.8 TOP: 6-7 Problem 3 Classifying a Quadrilateral KEY: coordinate plane graphing parallelogram opposite sides multi-part question DOK: DOK 3 3

9 9. ANS: Answers may vary. Sample: PTS: 1 DIF: L2 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 1 Naming Coordinates KEY: kite algebra coordinate plane 10. ANS: Answers may vary. Sample: PTS: 1 DIF: L3 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 1 Naming Coordinates KEY: rhombus algebra coordinate plane 11. ANS: Each diagonal has length (a b) 2 c 2. PTS: 1 DIF: L4 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 2 Using Variable Coordinates KEY: algebra coordinate plane isosceles trapezoid trapezoid diagonal 4

10 12. ANS: PTS: 1 DIF: L3 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 2 Using Variable Coordinates KEY: algebra coordinate plane square ESSAY 13. ANS: [4] Shows ABCD is a parallelogram (by any of several methods); then shows diagonals are perpendicular by computing slopes to be 3 2 and 2. Includes meaningful commentary 3 on what is occurring. [3] Shows ABCD is a parallelogram and shows diagonals are perpendicular, but presentation is not clear. [2] work complete and shows correct ideas, but contains errors [1] work incomplete, but shows some understanding of what to do PTS: 1 DIF: L3 REF: 6-7 Polygons in the Coordinate Plane OBJ: Classify polygons in the coordinate plane STA: MA.912.G.1.1 MA.912.G.2.6 MA.912.G.3.1 MA.912.G.3.3 MA.912.G.4.1 MA.912.G.4.8 TOP: 6-7 Problem 2 Classifying a Parallelogram KEY: extended response rubric-based question reasoning writing in math rhombus 5

11 14. ANS: [4] midpoint of AB ( 3, 3) midpoint of BC (3, 3) midpoint of CD (3, 1) midpoint of DA ( 3, 1) The figure is a rectangle. [3] Shows correct midpoints and shape, but presentation is not clear. [2] work complete and shows correct ideas, but contains errors [1] work incomplete, but shows some understanding of what to do PTS: 1 DIF: L3 REF: 6-7 Polygons in the Coordinate Plane OBJ: Classify polygons in the coordinate plane STA: MA.912.G.1.1 MA.912.G.2.6 MA.912.G.3.1 MA.912.G.3.3 MA.912.G.4.1 MA.912.G.4.8 TOP: 6-7 Problem 3 Classifying a Quadrilateral KEY: extended response rubric-based question reasoning writing in math rhombus square rectangle 6

12 15. ANS: [4] Proofs may vary. Sample: For rhombus in the coordinate plane, as shown, the quadrilateral determined by the midpoints (a, b), ( a, b), (a, b), and ( a, b) has one pair of opposite sides vertical (no slope) and the other pair horizontal (slope 0), so the quadrilateral is a parallelogram with perpendicular sides, or a rectangle. [3] shows good setup and idea for proof, but has some small inaccuracies [2] shows reasonable setup and idea for proof, but has significant math difficulties [1] shows reasonable setup for proof PTS: 1 DIF: L4 REF: 6-9 Proofs Using Coordinate Geometry OBJ: Prove theorems using figures in the coordinate plane TOP: 6-9 Problem 1 Writing a Coordinate Proof KEY: rhombus midpoint rectangle extended response rubric-based question coordinate plane algebra writing in math reasoning DOK: DOK 3 7

13 16. ANS: [4] Proofs may vary. Sample: Given: Line l is the perpendicular bisector of CD. Prove: Point R(a, b) is equidistant from points C and D. By the Distance Formula, CR (a 0) 2 (b 0) 2 a 2 b 2 DR (a 2a) 2 (b 0) 2 a 2 b 2 Because CR DR, point R on the perpendicular bisector of the segment is equidistant from the endpoints of the segment. [3] shows good setup and idea for proof, but has some small inaccuracies [2] shows reasonable setup and idea for proof, but has significant math difficulties [1] shows reasonable setup for proof PTS: 1 DIF: L4 REF: 6-9 Proofs Using Coordinate Geometry OBJ: Prove theorems using figures in the coordinate plane TOP: 6-9 Problem 1 Writing a Coordinate Proof KEY: rhombus midpoint rectangle extended response rubric-based question coordinate plane algebra writing in math reasoning DOK: DOK 3 8

14 17. ANS: [4] Proofs may vary. Sample: Answers may vary. Sample: Given: WY and XZ are diagonals of rectangle WXYZ. Prove: WY XZ Distance of XZ (a 0) 2 (0 b) 2 a 2 b 2 WY (a 0) 2 (b 0) 2 a 2 b 2 By the definition of congruency, diagonals XZ and WY of rectangle WXYZ are congruent. [3] shows good setup and idea for proof, but has some small inaccuracies [2] shows reasonable setup and idea for proof, but has significant math difficulties [1] shows reasonable setup for proof PTS: 1 DIF: L4 REF: 6-9 Proofs Using Coordinate Geometry OBJ: Prove theorems using figures in the coordinate plane TOP: 6-9 Problem 2 Writing a Coordinate Proof KEY: rhombus midpoint rectangle extended response rubric-based question coordinate plane algebra writing in math reasoning DOK: DOK 3 9

15 OTHER 18. ANS: Answers may vary. Sample: JKL is scalene. All three sides have different lengths. PTS: 1 DIF: L3 REF: 6-7 Polygons in the Coordinate Plane OBJ: Classify polygons in the coordinate plane STA: MA.912.G.1.1 MA.912.G.2.6 MA.912.G.3.1 MA.912.G.3.3 MA.912.G.4.1 MA.912.G.4.8 TOP: 6-7 Problem 1 Classifying a Triangle KEY: scalene isosceles triangle distance formula 10

16 19. ANS: Answers may vary. Sample: slope of AB is 7 6 slope of BC is 6 7 slope of CD is 7 6 slope of AD is 6 7 AB CD and BC AD, so ABCD is a parallelogram. AB BC, BC CD, CD AD, and AB AD. ABC, BCD, CDA, BAD are right angles. ABCD is a rectangle. PTS: 1 DIF: L4 REF: 6-7 Polygons in the Coordinate Plane OBJ: Classify polygons in the coordinate plane STA: MA.912.G.1.1 MA.912.G.2.6 MA.912.G.3.1 MA.912.G.3.3 MA.912.G.4.1 MA.912.G.4.8 TOP: 6-7 Problem 2 Classifying a Parallelogram KEY: coordinate plane proof reasoning rectangle slope multi-part question 20. ANS: Answers may vary. Sample: Using a factor of 2 in each coordinate simplifies what you find for the coordinates of the midpoint of AB, namely (a + c, b + d). PTS: 1 DIF: L3 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 1 Naming Coordinates KEY: algebra coordinate plane graphing reasoning writing in math 11

17 21. ANS: Answers may vary. Sample: PTS: 1 DIF: L3 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 3 Planning a Coordinate Proof KEY: diagonal parallelogram algebra coordinate plane writing in math reasoning DOK: DOK ANS: Answers may vary. Sample: Given: FH and GK are diagonals of square FGHK. Prove: FH GK PTS: 1 DIF: L3 REF: 6-8 Applying Coordinate Geometry TOP: 6-8 Problem 3 Planning a Coordinate Proof KEY: diagonal parallelogram algebra coordinate plane writing in math reasoning DOK: DOK 3 12

18 23. ANS: 4] Proofs may vary. Sample: Given: AC and BD are diagonals of kite ABCD. Prove: AC BD Slope of DB 3b 3b 2a 0 0 Slope of AC 4b 0 a a 4b 0 = undefined A line with a zero slope is perpendicular to a line with an undefined slope, so the diagonals of the kite are perpendicular. [3] shows good setup and idea for proof, but has some small inaccuracies [2] shows reasonable setup and idea for proof, but has significant math difficulties [1] shows reasonable setup for proof PTS: 1 DIF: L3 REF: 6-9 Proofs Using Coordinate Geometry OBJ: Prove theorems using figures in the coordinate plane TOP: 6-9 Problem 2 Writing a Coordinate Proof KEY: diagonal kite algebra coordinate plane writing in math reasoning DOK: DOK 3 13

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