Name:: ______________________ ________________________ Period: __________
Ch 4 Review 2016-2017 1. In the diagram, name the vector and select its component form.
→
a.
→
c.
AO , −4,−1 →
b.
a. b.
d.
OA , 4,1
2. Write a rule for the translation of
ÊÁ x,y ˆ˜ → ÊÁ x − 1, y − 3 ˆ˜ Ë ¯ Ë ¯ ÊÁ x,y ˆ˜ → ÊÁ x + 2, y ˆ˜ Ë ¯ Ë ¯
AO , −1,−4 →
ABC to
OA , 1,4
A′B ′C ′ .
c. d.
ÊÁ x,y ˆ˜ → ÊÁ x, y + 2 ˆ˜ Ë ¯ Ë ¯ ÊÁ x,y ˆ˜ → ÊÁ x + 3, y + 2 ˆ˜ Ë ¯ Ë ¯
3. An architect is designing a garden for a client on a planning sheet. The client requests the relocation of a water fountain by the rule ÁÊË x, y ˜ˆ¯ → ÁÊË x + 9, y − 8 ˜ˆ¯ . Later, the client decides that they would prefer the water fountain’s location moved from this new position 5 units left and 3 units down. Find the composition as a single transformation. a. ÊÁË x,y ˆ˜¯ → ÊÁË x + 6, y − 13 ˆ˜¯ c. ÊÁË x, y ˆ˜¯ → ÊÁË x − 3, y − 5 ˆ˜¯ b. ÊÁË x, y ˆ˜¯ → ÊÁË x + 4, y − 11 ˆ˜¯ d. ÊÁË x, y ˆ˜¯ → ÊÁË x + 14, y − 5 ˆ˜¯
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4. The logo for a business is moved across a page 6 units right and 6 units down. Next, it is moved 2 units left and 2 units up. Rewrite the composition as a single translation. c. ÊÁË x, y ˆ˜¯ → ÊÁË x + 8, y − 8 ˆ˜¯ a. ÊÁË x, y ˆ˜¯ → ÊÁË x − 4, y + 4 ˆ˜¯ b. ÊÁË x, y ˆ˜¯ → ÊÁË x + 4, y − 4 ˆ˜¯ d. ÊÁË x, y ˆ˜¯ → ÊÁË x − 8, y + 8 ˆ˜¯ 5. Graph the polygon and its image after a rotation of 90° counterclockwise about the origin. a.
c.
b.
d.
6. The minute hand on a clock rotates in clockwise direction. If the clock was laid on a coordinate plane with its center at the origin and the minute hand currently had an endpoint of ÊÁË −4,−6 ˆ˜¯ , what will the coordinates of the endpoint be in 15 minutes? a. ÊÁË −6,4 ˆ˜¯ b. ÊÁË 6,4 ˆ˜¯
c. d.
ÊÁ 6,−4 ˆ˜ Ë ¯ ÊÁ 4,6 ˆ˜ Ë ¯
2
7. Graph the polygon and its image after a rotation of 180° about the origin. a.
c.
b.
d.
8. Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement.
a. b.
5 ; enlargement 2 2 ; reduction 3
c. d.
2 ; reduction 5 3 ; enlargement 2
3
9. Graph ST with endpoints at S ÊÁË 2,0 ˆ˜¯ and T ÊÁË −1, −3 ˆ˜¯ and its image after the composition. Translation: ÊÁË x, y ˆ˜¯ → ÊÁË x − 2, y + 2 ˆ˜¯ Rotation: 90° counterclockwise about the origin a.
c.
b.
d.
10. Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement.
a. b.
2 ; reduction 5 5 ; enlargement 2
c. d.
2 ; reduction 3 3 ; enlargement 2
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11. An image of a celebrity is going to be used on a promotional billboard. The person is 6.25 feet tall in real life, and 21.875 feet tall on the billboard. What is the scale factor of this dilation? a. 0.29 c. 14.0625 b. 15.625 d. 3.5 12. The person in line in front of you at the store is buying reading glasses that say + 1.25. If that is the magnification level of the glasses, how tall, to the nearest thousandth of an inch, will 13-point font (which is about 0.181 inch) appear? a. about 0.226 in. c. about 1.431 in. b. about 2.353 in. d. about 0.407 in. 13. A copy machine is set to enlarge an image 160%. If the original image is 3 inches tall, how tall will the enlarged image be? a. about 163 in. c. about 4.8 in. b. about 4.6 in. d. about 480 in. 14. You want to enlarge a picture by a factor of 4 from its current size of 5 inches by 7 inches. What is the size of the enlarged picture? a. 9 in. by 11 in. c. 20 in. by 28 in. b. 9 in. by 28 in. d. 20 in. by 11 in. 15. Select all possible endpoints for A′B ′ after a rotation about the origin is applied to AB with endpoints at A ÊÁË −8,1 ˆ˜¯ and B ÁÊË −6,−8 ˜ˆ¯ . c. A′ ÁÊË 1,8 ˜ˆ¯ ,B ′ ÁÊË −8,6 ˜ˆ¯ a. A′ ÁÊË −8,−1 ˜ˆ¯ ,B ′ ÁÊË −6,8 ˜ˆ¯ b. A′ ÁÊË 1,−8 ˜ˆ¯ ,B ′ ÁÊË −8,−6 ˜ˆ¯ d. A′ ÁÊË −1,−8 ˜ˆ¯ ,B ′ ÁÊË 8,−6 ˜ˆ¯ 16. The vertices of
CDE are C(–3, 3), D(4, 9), and E(6, 1). Translate
CDE using vector 5, − 3 .
17. Graph quadrilateral ABCD with vertices A(2, 8), B(6, 14), C(12, 20) and D(14, 10) and its image after the translation ÊÁË x, y ˆ˜¯ → ÊÁË x + 2, y − 2 ˆ˜¯ . Next 3 Questions: Match the coordinates of given that A ÊÁË −5,1 ˆ˜¯ and B ÊÁË −5,3 ˆ˜¯ . a. A′ ÊÁË 5,1 ˆ˜¯ , B ′ ÊÁË 5,3 ˆ˜¯ b. A′ ÊÁË −1,3 ˆ˜¯ , B ′ ÊÁË −1,5 ˆ˜¯ c. A′ ÊÁË −5,9 ˆ˜¯ ,B ′ ÊÁË −5,11 ˆ˜¯
A′B ′ to the transformation applied to AB so that AB ≅ A′B ′ A′ ÊÁË 5, −1 ˆ˜¯ , B ′ ÊÁË 5, −3 ˆ˜¯ e. A′ ÊÁË −1, −5 ˆ˜¯ , B ′ ÊÁË −3, −5 ˆ˜¯ f. A′ ÊÁË −1,5 ˆ˜¯ ,B ′ ÊÁË −3,5 ˆ˜¯ d.
18. reflection over the y-axis 19. rotation of 180° about the origin 20. reflection over the line y = x followed by a reflection over the line x = 0
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21. Graph
JKL with vertices J(1, 1), K(3, 5) and L(5, 3) and its image after the composition.
Translation: ÊÁË x, y ˆ˜¯ → ÊÁË x + 5, y − 2 ˆ˜¯ Translation: ÊÁË x, y ˆ˜¯ → ÊÁË x − 1, y + 2 ˆ˜¯ 22. Graph
BCD with vertices B(–1, 1), C(2, 3) and D(5, 1) and its image after the reflection in the line n: y = −1.
23. Find A’, B’, and C’ and graph the polygon and its image after the reflection in the line y = x .
24. Draw a 100° rotation of quadrilateral WXYP about point P.
25. Write the vector sum for 〈1, 6〉 and 〈−5, −2〉 a. 〈7, −7〉 b. 〈−4, 4〉
c.
〈−7, 7〉
d.
〈4, −4〉
26. Graph ST with S ÊÁË −3,0 ˆ˜¯ and T ÊÁË 1,−1 ˆ˜¯ and its image after the composition. First find S’, T’, S”, and T”. Translation: ÊÁË x, y ˆ˜¯ → ÊÁË x − 2, y + 2 ˆ˜¯ Rotation: 180° about the origin
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27. Graph
RST with R ÁÊË 1,3 ˜ˆ¯ , S ÁÊË −1, −2 ˜ˆ¯ and T ÁÊË 4,0 ˜ˆ¯ and its image after the composition. First find R”, S” and T”.
Reflection: in the x-axis Rotation: 90° counterclockwise about the origin 28. Graph ST with endpoints at S ÊÁË −4,4 ˆ˜¯ and T ÊÁË 3,0 ˆ˜¯ and its image after the composition. First find S” and T”. Rotation: 180° about the origin Reflection: in the line y = x
ABC with vertices A ÊÁË −2,4 ˆ˜¯ , B ÊÁË 0, −4 ˆ˜¯ , and C ÊÁË 2,0 ˆ˜¯ and its image after a dilation centered at (0, 0) with a scale factor of 2. First, find A’, B’, C’.
29. Graph
30. Graph quadrilateral ABCD with vertices A ÊÁË 0,6 ˆ˜¯ , B ÊÁË −3,−1.5 ˆ˜¯ , C ÊÁË 3,−4.5 ˆ˜¯ , and D ÊÁË 6,6 ˆ˜¯ and its image after a 2 dilation centered at (0, 0) with a scale factor of . First find A’, B’, C’, D’. 3
ABC with vertices A ÊÁË 0,4 ˆ˜¯ , B ÊÁË −2,−3 ˆ˜¯ , and C ÊÁË 2,2 ˆ˜¯ and its image after a dilation centered at (0, 0) with a scale factor of –2. First find A’, B’, C’.
31. Graph
ABC with vertices A ÊÁË 6,−3 ˆ˜¯ , B ÊÁË −6,12ˆ˜¯ , and C ÊÁË 0,−12 ˆ˜¯ and its image after a dilation centered at (0, 0) 1 with a scale factor of − . First find A’, B’, C’. 3
32. Graph
ABC with vertices A ÊÁË −2,3 ˆ˜¯ , B ÊÁË 0,−4 ˆ˜¯ , and C ÊÁË 2,0 ˆ˜¯ and its image after the similarity transformation. First find A”, B”, C”.
33. Graph
Translation: ÊÁË x, y ˆ˜¯ → ÊÁË x + 3, y − 2 ˆ˜¯ Dilation: ÊÁË x, y ˆ˜¯ → ÊÁË 1.5x, 1.5y ˆ˜¯
ABC with vertices A ÁÊË 2,4 ˜ˆ¯ , B ÁÊË −2,−1 ˜ˆ¯ , and C ÁÊË 4,1 ˜ˆ¯ and its image after the similarity transformation. First find A”, B”, C”.
34. Graph
Dilation: ÊÁË x, y ˆ˜¯ → ÊÁË 2.5x, 2.5y ˆ˜¯ Reflection: in the x-axis
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ABC with vertices A ÁÊË 10,5 ˜ˆ¯ , B ÁÊË 4,7 ˜ˆ¯ , and C ÁÊË 6,−4 ˜ˆ¯ and its image after the similarity transformation. First find A”, B”, C”.
35. Graph
Rotation: 90° counterclockwise about the origin ÊÁ 1 1 ˆ˜ Dilation: ÊÁË x, y ˆ˜¯ → ÁÁÁÁ x, y ˜˜˜˜ Ë2 2 ¯ 36. A car begins at point A. During its trip, the car moves in a straight line until it reaches a point C that is 3 miles south and 4 miles east relative to point A. The car then immediately moves to its final destination which is 2 miles south and 4 miles east relative to point C. The entire trip takes 19 minutes.
a. Write the transformation as a single composition using arrow notation.. b. Let the car’s final destination be point A’. What is the straight line distance from the car’s starting point to point A’ ? c. Find the car’s average speed throughout the entire trip in miles per hour. 37. You want to enlarge photos for a science class display. You experiment with the size using a photo that is 3 inches tall. First, you enlarge it by 365% but decide that the image is too tall. So, you use the new image and reduce it by 60%. Again, you are not really happy so you enlarge that image by 155%. You like the results. a. How tall is your image to the nearest tenth on an inch? b. What scale factor should you use on the rest of the pictures so you can enlarge them in one dilation and have them all stay proportional? 38. Two houses with coordinates A(0, -3) and B(6, 3) must connect to a waterline located at the line y = 6. What are the coordinates of the conection point C to minimize the amount of pipe used?
.
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39. HONORS: AB has endpoints at A ÊÁË −3,3 ˆ˜¯ and B ÊÁË 2,0 ˆ˜¯ . If AB is rotated 270° counterclockwise about the point ÊÁ 3,2 ˆ˜ , what are the coordinates of A′B ′? Ë ¯ 40. HONORS: Field scientists will be air dropped to a single location in the wilderness. From this location Group 1 must proceed to the coordinates A(4, 2) and Group 2 must proceed to the coordinates B(6, 4). a. The safest place to drop the scientists is on the bank of a river traveling east-west. If the river is the x-axis of their map, at what coordinates should the scientists be dropped to minimize the distance both groups have to travel? b. One of the two groups must proceed to a location C(1, –12) after reaching their assigned coordinates. Assuming the groups travel at the same speed, which group would reach this location first?
ABC with vertices A ÊÁË 4,6 ˆ˜¯ , B ÊÁË 5,7 ˆ˜¯ , and C ÊÁË −3,−9 ˆ˜¯ find the image coordinates after a dilation centered at (3, -2) with a scale factor of 2. Find A’, B’, C’.
41. HONORS: Given
42. HONORS: Given triangle ABC with A(2,0), B(0,0) and C(0,2). After a scale factor of k=3 is applied, find the perimeter of triangle A`B`C`. Round to the neartest tenth. a. 2.8 c. 8.5 b. 12.5 d. 20.5 43. HONORS: Given triangle ABC with A(2,0), B(0,0) and C(0,2). After a scale factor of k=3 is applied, find the area of triangle A`B`C`. a. 6 c. 18 b. 8 d. 36 44. HONORS: The point G(4, 8) is rotated 90° about point M(−7, − 9) and then reflected across the line y = −6 . Find the coordinates of the image G ′. a. (−24, − 14) c. (−8, − 16) b. (12, − 14) d. (−18, − 20) 45. HONORS: The equation of the line a is y = −2x + 5 . Find the equation of the image of line a after a dilation centered at the origin with scale factor 2. a. y = −4x + 10 c. y = −2x + 10 5 b. y = −2x + d. 2y = −4x + 10 2
9
ID: A
Ch 4 Review 2016-2017 Answer Section 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
D B B B C A A C A A D A C C C, D C'(2, 0), D'(9, 6), E'(11, –2)
17. 18. A 19. D
1
ID: A 20. E
21.
22.
23.
2
ID: A
24. 25. B
26.
S ′ ÁÊË −5,2 ˜ˆ¯ ,T ′ ÁÊË −1,1 ˜ˆ¯ S ″ ÁÊË 5,−2˜ˆ¯ ,T ″ ÁÊË 1,−1 ˜ˆ¯
27.
R ′ ÊÁË 1,−3 ˆ˜¯ ,S ′ ÊÁË −1,2 ˆ˜¯ ,T ′ ÊÁË 4,0 ˆ˜¯ R ″ ÊÁË 3,1 ˆ˜¯ ,S ″ ÊÁË −2,−1 ˆ˜¯ ,T ″ ÊÁË 0,4 ˆ˜¯ 3
ID: A
28.
S ′ ÊÁË 4,−4 ˆ˜¯ ,T ′ ÊÁË −3,0 ˆ˜¯ S ″ ÊÁË −4,4ˆ˜¯ ,T ″ ÊÁË 0,−3 ˆ˜¯
29.
A′ ÊÁË −4,8 ˆ˜¯ ,B ′ ÊÁË 0,−8 ˆ˜¯ ,C ′ ÊÁË 4,0 ˆ˜¯
4
ID: A
30.
A′ ÁÊË 0,4 ˆ˜¯ , B ′ ÁÊË −2,−1 ˜ˆ¯ , C ′ ÁÊË 2,−3 ˜ˆ¯ , D ′ ÁÊË 4,4 ˜ˆ¯
31.
A′ ÁÊË 0,−8 ˆ˜¯ ,B ′ ÁÊË 4,6 ˜ˆ¯ ,C ′ ÁÊË −4,−4 ˜ˆ¯
5
ID: A
32.
A′ ÁÊË −2,1 ˆ˜¯ , B ′ ÁÊË 2,−4 ˜ˆ¯ , C ′ ÁÊË 0,4 ˜ˆ¯
33.
A′ ÊÁË 1,1 ˆ˜¯ ,B ′ ÊÁË 3,−6 ˆ˜¯ ,C ′ ÊÁË 5,−2 ˆ˜¯ A″ ÊÁË 1.5,1.5 ˆ˜¯ ,B ″ ÊÁË 4.5,−9 ˆ˜¯ ,C ″ ÊÁË 7.5,−3 ˆ˜¯
6
ID: A
34.
A′ ÊÁË 5,10 ˆ˜¯ ,B ′ ÊÁË −5,−2.5 ˆ˜¯ ,C ′ ÊÁË 10,2.5 ˆ˜¯ A″ ÊÁË 5,−10 ˆ˜¯ ,B ″ ÊÁË −5,2.5 ˆ˜¯ ,C ″ ÊÁË 10,−2.5 ˆ˜¯
35.
A′ ÊÁË −5,10 ˆ˜¯ ,B ′ ÊÁË −7,4 ˆ˜¯ ,C ′ ÊÁË 4,6 ˆ˜¯ A″ ÊÁË −2.5,5 ˆ˜¯ ,B ″ ÊÁË −3.5,2 ˆ˜¯ ,C ″ ÊÁË 2,3 ˆ˜¯ 36. a. ÊÁË x,y ˆ˜¯ → ÊÁË x + 8,y − 5 ˆ˜¯ b. about 9.43 mi c. about 29.91 mi/h 37. a. about 10.2 in. b. about 339% 38. (4.5, 6)
7
ID: A 39. A′ ÁÊË 4,8 ˜ˆ¯ , B ′ ÁÊË 1,3 ˜ˆ¯
ÊÁ 14 ˆ˜ 40. a. ÁÁÁÁ ,0 ˜˜˜˜ Ë 3 ¯ b. Group 1 41. A’(5,14), B’(7, 16), C’(-9, -16) 42. D 43. C 44. A 45. C
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