116 Chapter 6 Transformations and the Coordinate Plane


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1 116 Chapter 6 Transformations and the Coordinate Plane Chapter 61 The Coordinates of a Point in a Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. If the coordinates of ABC are A(1, 1), B(6, 1), and C(1, 7), then the area of the polgon is (1) 15 (2) 18 (3) 30 (4) If the coordinates of quadrilateral ABCD are A( 1, 0), B(5, 0), C(5, 4), and D( 1, 4), then the area of the quadrilateral is (1) 12 (2) 20 (3) 24 (4) If two of the coordinates of PQR are P(6, 3) and Q(6, 3) and the area of PQR is 24, then which of the following could be the coordinate of point R? (1) ( 2, 0) (2) ( 1, 0) (3) (2, 0) (4) (8, 0) 4. In the point (a, b), a 2 and b 2. In which quadrant does the point lie? (1) Quadrant I (2) Quadrant II (3) Quadrant III (4) Quadrant IV 5. If (a, b) is a point in quadrant II, then ( a, b) lies in quadrant (1) I (2) II (3) III (4) IV 6. If 0 and 0, then the coordinates of point P(, ) lies in quadrant (1) I (2) II (3) III (4) IV
2 Chapter 6 Transformations and the Coordinate Plane 117 PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 7. a. Graph points A( 1, 4) and B( 1, 3). b. If AB is a side of square ABCD, what are the possible coordinates of C and D? (Hint: There will be two answers.) Answer: (6, 4) and (6, 3) or ( 8, 4) and ( 8, 3) c. Find the area of ABCD. Answer: 49 square units Solution: AB 3 ( 4) 7 AB The coordinates of ABC are A( 2, 0), B(0, 5), and C(2, 0). a. Graph ABC. b. What kind of triangle is ABC? Answer: Isosceles c. Use congruent triangles to justif our answer to part b. Eplanation: Let the midpoint of side AC be D. Then, AD > CD. The coordinates of D are (0, 0), so the median BD is on the ais. Since AC is on the ais, BD ' AC and angles BDA and BDC are congruent right angles. BD > BD b the refleive propert, so BDA BDC b SAS. Corresponding parts of congruent triangles are congruent, so AB > CB and ABC is isosceles B A B A C
3 118 Chapter 6 Transformations and the Coordinate Plane Chapter 62 Line Reflections Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. If ABC has line smmetr onl when the median is drawn from verte B, what kind of triangle is ABC? (1) right (3) isosceles (2) scalene (4) equilateral 2. A line reflection preserves (1) distance and orientation. (2) angle measure and orientation. (3) distance but not angle measure. (4) distance and angle measure. 3. What kind of smmetr does the name TT have? (1) vertical line smmetr onl (2) horizontal line smmetr onl (3) both vertical and horizontal line smmetr (4) neither horizontal nor vertical line smmetr 4. Which letter has both vertical and horizontal line smmetr? (1) A (3) T (2) M (4) X 5. If ABC has line smmetr when an altitude is drawn from verte A or B, what kind of triangle is ABC? (1) right (2) isosceles (3) equilateral (4) scalene 6. In the given figure, m and k are smmetr lines for the regular pentagon ABCDE. A E B k Find E, the image of E under r m followed b r k. (1) A (3) C (2) B (4) D D C m
4 Chapter 6 Transformations and the Coordinate Plane 119 PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] In 7 8, refer to the following diagram. B C m t A D 7. In the given figure, m and t are smmetr lines for quadrilateral ABCD. Find the image of each point under the given reflection(s). a. r t (B) b. r m (A) Answer: A Answer: A c. The image of C under r t followed b r m Answer: B Solution: r t (C) D r m (D) B d. The image of A under r t followed b r m followed b r t Answer: C Solution: r t (A) B r m (B) D r t (D) C 8. Find the image of each segment under the given reflection(s). a. r m (BC) b. r t (AB) Answer: CD Answer: AB c. The image of AB under r m followed b r t Answer: BC Solution: d. The image of CD under r m followed b r t followed b r m Answer: AB Solution: r m (AB) 5 AD r t (AD) 5 BC r m (CD) 5 BC r t (BC) 5 AD r m (AD) 5 AB
5 120 Chapter 6 Transformations and the Coordinate Plane Chapter 63 Line Reflections in the Coordinate Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. What are the coordinates of B, the image of B(2, 1) after a reflection in the line? (1) ( 2, 1) (2) ( 1, 2) (3) (1, 2) (4) (2, 1) 2. What is the image of ( 4, 3) when reflected in the ais? (1) ( 4, 3) (2) ( 3, 4) (3) (3, 4) (4) (4, 3) 3. The coordinates of P, the reflection of P(2, 6) in the line, are (1) ( 6, 2) (2) ( 6, 2) (3) (2, 6) (4) (6, 2) 4. When the points A( 5, 0), B(0, 5), C(5, 0), and D(0, 5) are joined, square ABCD is formed. Which of the following is not a line of smmetr? (1) 0 (2) 5 (3) (4) 5. If P, the image of point P under a reflection in the ais, is in the second quadrant, in what quadrant is P, the image of P under a reflection in the line? (1) I (2) II (3) III (4) IV 6. If point A has coordinates (4, 5), what are the coordinates of A, the image of A when reflected first across 6 and then reflected across 2? (1) ( 4, 5) (2) (5, 3) (3) (5, 13) (4) (12, 5)
6 Chapter 6 Transformations and the Coordinate Plane 121 PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 7. The coordinates of the vertices of a rectangle are (1, 3), (7, 3), (7, 1), and (1, 1). a. Write the equation of the vertical line of smmetr of the rectangle. Answer: 4 b. Write the equation of the horizontal line of smmetr of the rectangle. Answer: 1 c. Find the area of the rectangle. Answer: 24 square units 8. Triangle PQR has coordinates P( 1, 3), Q(3, 7), and R(0, 6). a. State the coordinates of P Q R, the image of PQR under a reflection in the line. Answer: P (3, 1), Q (7, 3), R (6, 0) b. State the coordinates of P Q R, the image of P Q R under a reflection in the ais. Answer: P ( 3, 1), Q ( 7, 3), R ( 6, 0)
7 122 Chapter 6 Transformations and the Coordinate Plane Chapter 64 Point Reflections in the Coordinate Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. If the point ( 3, 2) is reflected in the origin, the coordinates of its image are (1) ( 3, 2) (3) (2, 3) (2) ( 2, 3) (4) (3, 2) 2. Which of the following graphs has point smmetr? (1) (2) 3. Point A (3, 5) is the reflection of point A( 1, 3). Which of the following is the point of reflection? (1) (2, 4) (2) (1, 4) (3) (1, 2) (4) (0, 0) 4. Which figure has no point smmetr? (1) square (2) rhombus (3) equilateral triangle (4) isosceles triangle 5. Which capital letter does not have point smmetr? (3) (1) S (3) C (2) N (4) H 6. Under a reflection in the origin, the image of (, ) is (4) (1) (, ) (2) (, ) (3) (, ) (4) (, )
8 Chapter 6 Transformations and the Coordinate Plane 123 PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 7. a. What is the image of P(3, 5) under a reflection in the origin? Answer: ( 3, 5) b. What is the image of Q(4, 0) under a reflection in the origin? Answer: ( 4, 0) c. Is the reflection of PQ in the ais the same as the reflection of PQ in the origin? Justif our answer. Answer: No Eplanation: Under a reflection in the ais, the image of P is ( 3, 5). 8. BAG with coordinates B(2, 1), A( 2, 3), and G(3, 5) is reflected through the origin. a. What are the coordinates of B A G, the image of BAG? Answer: B ( 2, 1), A (2, 3), G ( 3, 5) b. Graph both BAG and B A G. A B G G B A
9 124 Chapter 6 Transformations and the Coordinate Plane Chapter 65 Translations in the Coordinate Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. If the image of A(5, 3) under the transformation T, is (9, 1), find the image of B( 1, 2) under the same transformation. (1) ( 2, 3) (3) (3, 2) (2) (2, 3) (4) (3, 1) 2. If the image of D( 2, 5) under the translation T, is ( 5, 1), find the image of E( 3, 7) under the same translation. (1) ( 6, 3) (3) (0, 3) (2) ( 6, 4) (4) (0, 3) 3. If A( 2, 3) is transformed to B(3, 2) b T a,b and P(4, 3) is transformed to Q b T b,a, then what are the coordinates of Q? (1) (2, 7) (3) (3, 2) (2) (3, 2) (4) (9, 4) 4. If a translation maps point P( 1, 5) onto point P (5, 1), what is the translation? (1) T 6,26 (3) T 4,4 (2) T 26,6 (4) T 4,24 5. If the transformation T, maps point M( 5, 8) onto point M (1, 3), what is the value of? (1) 6 (3) 5 (2) 5 (4) 6 6. Given P(0, 2) and Q(5, 0), which translation maps P onto M(2.5, 1), the midpoint of PQ? (1) T 2.5,21 (3) T 22.5,21 (2) T 2.5,1 (4) T 22.5,1 PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 7. ABC has coordinates A( 1, 3), B(3, 7), and C(0, 6). a. A translation maps P(0, 0) onto P (0, 5). State the coordinates of the vertices of A B C, the image of ABC under the same transformation. Answer: A ( 1, 2), B (3, 2), C (0, 1) b. State the coordinates of the vertices of A B C, the image of A B C after a reflection through the origin. Answer: A (1, 2), B ( 3, 2), C (0, 1) 8. ABC has coordinates A(2, 3), B(4, 0), and C( 3, 2). a. Under the translation P(, ) P ( 4, 3), the image of ABC is A B C. State the coordinates of the vertices of A B C. Answer: A (6, 0), B (8, 3), C (1, 5) b. What are the coordinates of the vertices of A B C, the image of A B C under the translation? Answer: A ( 1, 2), B (1, 1), C ( 6, 3) T 27,2
10 Chapter 6 Transformations and the Coordinate Plane 125 Chapter 66 Rotations in the Coordinate Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. Which letter has rotational smmetr? (1) E (3) K (2) T (4) H 4. In the figure given, square ABCD is inscribed in circle. B A D C In the given figure, each polgon is the image of the other under a rotation about a fied point. Which is most likel the coordinates of the center of rotation? (1) (0, 0) (2) ( 3, 1) (3) ( 4, 2) (4) ( 5, 1) 3. What is the image of A(2, 5) under R 90? (1) ( 5, 2) (2) ( 2, 5) (3) (2, 5) (4) (5, 2) What is R 90 (A)? (1) A (3) C (2) B (4) D 5. Which figure has 120 rotational smmetr? (1) square (2) rhombus (3) parallelogram (4) equilateral triangle 6. If B is the image of A(2, 3) after a 90 rotation clockwise about the origin, the coordinates of B are (1) (3, 2) (3) ( 2, 3) (2) ( 2, 3) (4) ( 3, 2)
11 126 Chapter 6 Transformations and the Coordinate Plane PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 7. Find the image of each point under a rotation about the origin of: a. 90 counterclockwise, b. 90 clockwise, and c. 180 counterclockwise. (1) (3, 2) Answer: a. (2, 3) b. ( 2, 3) c. ( 3, 2) (2) ( 5, 5) Answer: a. (5, 5) b. ( 5, 5) c. (5, 5) (3) ( 2, 6) Answer: a. ( 6, 2) b. (6, 2) c. (2, 6) 8. Given: Square ABCD with E, F, G, H the midpoints of the sides; AC, BD, EG, and FH intersect at point. B G C Find: a. R 908 (G) F H Answer: F A E D b. R 908 A HD B Answer: GC c. R 908 (/FA) Answer: ED d. R 1808 (/BG) Answer: DE
12 Chapter 6 Transformations and the Coordinate Plane 127 Chapter 67 Glide Reflections Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. Which propert is not preserved b a glide reflection? (1) betweenness (2) angle measure (3) orientation (4) collinearit 2. Find the image of point P(5, 4) under the glide reflection T 23,23 followed b r =. (1) (2, 1) (2) (1, 2) (3) ( 1, 2) (4) ( 12, 15) 3. A glide reflection is defined b r 521 followed b T a,b. If the image of P( 5, 2) is P (3, 8) under this transformation, what are the values of a and b? (1) a 10, b 0 (2) a 10, b 8 (3) a 0, b 10 (4) a 8, b Under a glide reflection, r =1 is followed b T a,b.if A(6, 4) A ( 7, 2) under this transformation, what are the values of a and b? (1) a 0, b 13 (2) a 6, b 13 (3) a 13, b 0 (4) a 13, b 6 5. Under a glide reflection, the image of A(4, 3) is A ( 4, 3) and the image of B( 7, 8) is B (7, 2). If the line of reflection is vertical, which of the following is the line of reflection? (1) 1 (2) 0 (3) 1 (4) 0 6. Under a glide reflection, the image of Q(0, 2) is Q (4, 2). If the translation is T 2,2, which of the following is the line of reflection? (1) 2 (2) 2 (3) (4)
13 128 Chapter 6 Transformations and the Coordinate Plane PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 7. Given: DE with D(2, 6) and E(5, 10) Graph and state the coordinates of the endpoints of: a. DrEr, the image of DE under the reflection r. Answer: D (6, 2), E (10, 5) b. DsEs, the image of DrEr after the translation T 3,3. Answer: D (9, 5), E (13, 8) D E E D Given: ABC with A( 6, 3), B( 1, 5), and C( 3, 8). Graph and state the coordinates of the vertices of: a. A B C, the image of ABC under the translation T 3,0. Answer: A ( 3, 3), B (2, 5), C (0, 8) b. A B C, the image of A B C under the reflection r =2. Answer: A ( 3, 1), B (2, 1), C (0, 4) C C A 3 2 A B B
14 Chapter 6 Transformations and the Coordinate Plane 129 Chapter 68 Dilations in the Coordinate Plane Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. What are the coordinates of P, the image of point P( 2, 1) under a dilation of 3 with respect to the origin? (1) ( 6, 3) (3) (3, 6) (2) ( 3, 6) (4) (6, 3) 2. If A (3, 1) is the image of point A(9, 3) under a dilation with respect to the origin, what is the rule for the dilation? (1) (3) D 2 D1 3 (2) D 1 (4) D 3 3. If D 2 (M) ( 2, 6), what are the coordinates of M? (1) ( 4, 4) (3) (4, 12) (2) (1, 3) (4) (4, 3) 4. Which propert is not preserved under dilation? (1) distance (3) angle measurement (2) midpoint (4) collinearit 5. Using the rule (, ) (4,4), find the coordinates of the image of the point ( 4, 12). (1) (16, 48) (3) ( 1, 3) (2) (1, 3) (4) ( 16, 48) 6. What are the coordinates of P, the image of P(4, 3) under T 2,1 followed b D1 2? (1) ( 1, 3) (3) (3, 2) (2) (3, 1) (4) (3, 2) PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 7. If ABC has vertices A(0, 6), B(6, 3), and C( 3, 6), write the coordinates of the vertices of A B C, the image of ABC under the composition of transformations r ais followed b D1 3. Answer: A (0, 2), B ( 2, 1), C (1, 2) Solution: r ais (0, 6) (0, 6) D1 3 (0, 26) (0, 2) r ais (6, 3) ( 6, 3) D1 3 (26, 3) ( 2, 1) r ais ( 3, 6) (3, 6) D1 3 (3, 6) (1, 2) 8. Each of the following transformations is a composition of a dilation and a reflection in either the ais or the ais. a. Write a rule for the composition of transformations for which the image of A is A, such that A(3, 7) A (6, 14). Answer: r ais followed b D 2 or D 2 followed b r ais b. Write a rule for the composition of transformations for which the image of B is B, such that B(10, 6) B ( 5, 3). Answer: r ais followed b D1 or D1 2 2 followed b r ais
15 130 Chapter 6 Transformations and the Coordinate Plane Chapter 69 Transformations as Functions Section Quiz [20 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [20] 1. Which transformation is a direct isometr? 6. (1) D 3 (3) D 22 (2) r ais (4) T 2,3 2. The graph of g() is shown. g() Which of the following graphs represents the image of the given triangle under the composition r 5 + R 908? (1) Which of the following graphs represents r ais + r ais (g())? (1) (3) (2) (2) (4) 3. What is the image of point (2, 2) under the composition r ais + R 908? (1) (2, 2) (3) ( 2, 2) (2) (2, 2) (4) ( 2, 2) (3) 4. If the coordinates of P are ( 3, 7), what are the coordinates of D 2 + r 5 (P)? (1) ( 14, 6) (3) (6, 14) (2) ( 6, 14) (4) (14, 6) 5. Which transformation is an opposite isometr? (1) dilation (3) rotation of 90 (2) line reflection (4) translation (4)
16 Chapter 6 Transformations and the Coordinate Plane 131 PART II Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 7. a. What are the coordinates of P, the image of P(2, 3) under the composition of transformations r ais + r 5? Answer: (3, 2) Solution: r = (2, 3) (3, 2) r ais (3, 2) (3, 2) b. What single translation maps P P? Answer: T 1, 5 c. What are the coordinates of P, the image of P(2, 3) under the composition of transformations r 5 + R 908? Answer: (2, 3) Solution: R 90 (2, 3) ( 3, 2) r = ( 3, 2) (2, 3) d. What single transformation maps P P? State all answers. Answer: r ais or T 0, 6 8. If is the center of regular octagon ABCDEFGH, find the result for the composition of functions operating on the given verte. a. r AE + R 908 (B) B C D Answer: B Solution: R 90 (B) H A E b. R r HD (E) Answer: G r AE (H) B H G F Solution: (E) C r HD R 180 (C) G c. d. r AE + R 908 (H) Answer: D Solution: R 90 (H) F r AE (F) D R R 2458 (F) Answer: C Solution: R 45 (F) G R 180 (G) C
17 132 Chapter 6 Transformations and the Coordinate Plane Chapter 6 Transformations and the Coordinate Plane Chapter Review [40 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [16] 1. Which word does not have horizontal line smmetr? (1) CDE (3) TT (2) BB (4) HIDE 2. Under a certain translation, the image of (3, 1) is ( 2, 2). What is the image of (1, 2) under the same translation? (1) ( 5, 2) (3) ( 2, 1) (2) ( 4, 1) (4) ( 1, 4) 3. If the point (0, 3) is rotated 90 clockwise about the origin, its image is on line (1) 0 (3) (2) 0 (4) 4. If point P( 3, 7) is reflected in the line 1, what are the coordinates of P? (1) ( 3, 8) (3) ( 3, 5) (2) ( 3, 6) (4) ( 3, 6) 5. A transformation maps (3, 5) onto ( 5, 3). This transformation is equivalent to a (1) rotation of 90. (2) reflection in the origin. (3) reflection in the line. (4) translation of ( 5, 3). 6. A is the image of point A( 3, 4) after a translation defined b T 7,21. What other transformation on A would produce A? (1) (3) R 90 r 5 (2) r ais (4) R Which transformation is not an isometr? (1) T (5, 3) (3) r ais (2) D 3 (4) R If P(2, 4) P ( 1, 2) under a dilation D k, then k equals 1 (1) 4 (3) 2 (2) 2 (4) PART II Answer all questions in this part. Each correct answer will receive 2 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [4] 9. In which quadrant does the image of (5, 7) lie under the translation that maps (, ) to ( 6, 3)? Answer: III 10. What single translation is equivalent to T 3,21 followed b T 25,5? Answer: T 2,4
18 Chapter 6 Transformations and the Coordinate Plane 133 PART III Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 11. Find the image of the point whose coordinates are (3, 2) under the given composition of transformations. a. R + r 5 Answer: (2, 3) Solution: r = (3, 2) ( 2, 3) R ( 2, 3) (2, 3) b. R T 21, 22 Answer: (4, 2) Solution: T 1, 2 (3, 2) (2, 4) R 90 (2, 4) (4, 2) c. r ais + r 52 Answer: ( 1, 2) Solution: r =2 (3, 2) (1, 2) r ais (1, 2) ( 1, 2) 12. Lines l and m are lines of smmetr for rhombus PQRS. Find each of the following: a. r m A PQ B Q m R Answer: PS b. R,1808 (R) Answer: P P S l c. r l + r m A QR B Answer: PS Solution: r m (QR) 5 RS r l (RS) 5 PS
19 134 Chapter 6 Transformations and the Coordinate Plane PART IV Answer all questions in this part. Each correct answer will receive 6 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [12] 13. The coordinates of the endpoints of PQ are P( 2, 4) and Q(4, 1). a. n a set of aes, graph PQ. b. n the same set of aes, graph and state the coordinates of PrQr, the image of PQ under a reflection in the ais. Answer: P ( 2, 4), Q (4, 1) c. n the same set of aes, graph and state the coordinates of PsQs, the image of PrQr under the translation that maps (, ) to ( 2, ). Answer: P (0, 4), Q (6, 1) d. Is PrQr > PsQs? Justif our answer. Answer: Yes. Eplanation: Translation preserves distance, so P Q P Q. B the definition of congruent segments, PrQr > PsQs. 7 6 P Q Q 5 7 Q 2 3 P 4 5 P Given: MAD has coordinates M( 3, 4), A(2, 9), and D( 1, 2). a. State the coordinates of the vertices of M A D, the image of MAD under the composition R r ais. Answer: M (4, 3), A (9, 2), D ( 2, 1) Solution: r ais ( 3, 4) ( 3, 4) R 90 ( 3, 4) (4, 3) r ais (2, 9) (2, 9) R 90 (2, 9) (9, 2) r ais ( 1, 2) ( 1, 2) R 90 ( 1, 2) ( 2, 1) b. What single translation transformation is equivalent to R r ais? Answer: r =
20 Chapter 6 Transformations and the Coordinate Plane 135 Chapter 6 Transformations and the Coordinate Plane Cumulative Review [40 points] PART I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [16] 1. If B is the midpoint of line segment AC and D is the midpoint of line segment AB, which statement is true? (1) AB CB (3) AC AB (2) AD BD (4) DC 3BD 2. In the given figure, l, m, and n are lines with l m. l n m 5. The triangle with vertices at (0, 0), (6, 0), and ( 2, 2) is (1) right (3) scalene (2) isosceles (4) equilateral 6. Which of the following is a direct isometr? (1) D 2 (3) r ais (2) r (4) R In the given figure, BED, CEA, B D, and E is the midpoint of BD. B C Which angles are complementar? E (1) 1 and 3 (3) 2 and 3 (2) 1 and 2 (4) 3 and 5 A D 3. AB h and AC h are opposite ras if and onl if (1) B and C are on the same side of A. (2) B is between A and C. (3) A is between B and C. (4) A, B, and C are collinear. 4. Which statement is the converse of If two sides of a triangle are congruent, then the triangle is isosceles? (1) If a triangle is not isosceles, then two sides of the triangle are not congruent. (2) If two sides of a triangle are not congruent, then the triangle is not isosceles. (3) If a triangle is isosceles, then two sides of the triangle are congruent. (4) If two sides of a triangle are congruent, then the triangle is not isosceles. Which postulate justifies BEC DEA? (1) SSS (3) SSA (2) ASA (4) SAS 8. If the point (6, 0) is mapped onto point (6, 3) b some line reflection, we could describe the line of reflection as (1) the ais. (2) parallel to the ais. (3) parallel to the ais. (4) the line.
21 136 Chapter 6 Transformations and the Coordinate Plane PART II Answer all questions in this part. Each correct answer will receive 2 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [4] 9. In the given figure, m ABC 6,m CBD 3 11, and m ABD 86. What is the value of? Answer: 25 Solution: A D B C In an isosceles right triangle, the measure of an acute angle is represented b 2 5. Find the value of. Answer: 25 Solution: PART III Answer all questions in this part. Each correct answer will receive 4 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [8] 11. Find the coordinates of A, the image of A(8, 2) under the given composition of transformations r ais + T 24,7 + R Answer: ( 12, 9) Solution: R 1808 (8, 22) 5 R R 908 (8, 22) 5 R 908 (2, 8) 5 (28, 2) T 4,7 ( 8, 2) ( 12, 9) r ais ( 12, 9) ( 12, 9)
22 Chapter 6 Transformations and the Coordinate Plane Given: PQ > QR and PT > RT Q Prove: QT bisects PQR. T Proof: Statements P Reasons R 1. PQ > QR and PT > RT 1. Given. 2. QT > QT 2. Refleive propert of congruence. 3. PQT RQT 3. SSS. 4. PQT RQT 4. Corresponding parts of congruent triangles are congruent. 5. QT bisects PQR. 5. Definition of angle bisector. PART IV Answer all questions in this part. Each correct answer will receive 6 credits. Clearl indicate the necessar steps, answer with no work shown will receive onl 1 credit. [12] 13. a. The measures of the acute angles of a right triangle are in the ratio 1 : 4. What is the number of degrees in the measure of each of the acute angles? Answer: 18 and 72 Solution: b. In isosceles triangle PQR, QP > QR,m Q 5 13, and m P 2 7. Find the measure of each angle in PQR. Answer: m Q 98, m P m R 41 Solution: (17) , (17)
23 138 Chapter 6 Transformations and the Coordinate Plane 14. Given: AE h and GF h, HC > GD, 1 2, and AB > CD. Prove: H G H G A B 1 C 3 D 2 E Proof: F Statements Reasons 1. HC > GD, 1 2, and AB > CD 1. Given Vertical angles are congruent Transitive propert of congruence. 4. AC 5 AB 1 BC, BD 5 BC 1 CD 4. Partition postulate. 5. AB 1 BC > BC 1 CD 5. Addition postulate. 6. AC > BD 6. Substitution postulate. 7. AHC BGD 7. SAS. 8. H G 8. Corresponding parts of congruent triangles are congruent.
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