1. Warm Up #33 Monday 5/16

2. Homework
Monday 5/16 – Reflections of Shapes page 3 and 4

3. Find the area and perimeter
Warm Up #34 Monday 5/17
The diagonal of a rectangle is 23 inches. The width is 20 inches. What is the area of the rectangle? Round your answer to the nearest tenths.
Find the area and perimeter

4. Homework
Tuesday 5/17– Reflections of Shapes finish

5. Reflections of Shapes on Coordinate Plane

6. Transformation – a change in the ___________, _________ , or _________ of a figure.
Rigid Transformation – a change in the position of a figure that does not change its _______________ or ________________.
Preimage – the __________ figure in a transformation.
Image – the _____________ figure in a transformation.
Isometry – a transformation in which the original figure and its image are ___________.
size
shape
position
size
shape
original
resulting
congruent

7. Transformations on the Coordinate Plane
Algebra 4-2 Transformations on the Coordinate Plane
Line of
reflection
NOT A RIGID TRANSFOMRATION
NOT AN ISOMETRY
Transformations on the Coordinate Plane
Harbour

8. Identify Transformations
Identify the transformation as a reflection, translation, dilation, or rotation.
Answer: The figure has been increased in size. This is a dilation.
Example 2-1a

9. Identify Transformations
Identify the transformation as a reflection, translation, dilation, or rotation.
Answer: The figure has been shifted horizontally to the right. This is a translation.
Example 2-1b

10. Identify Transformations
Identify the transformation as a reflection, translation, dilation, or rotation.
Answer: The figure has been turned around a point. This is a rotation.
Example 2-1c

11. Identify Transformations
Identify the transformation as a reflection, translation, dilation, or rotation.
Answer: The figure has been flipped over a line. This is a reflection.
Example 2-1d

12. Identify Transformations
Identify each transformation as a reflection, translation, dilation, or rotation. a. b. c. d.
Answer: rotation
Answer: reflection
Answer: dilation
Answer: translation
Example 2-1e

13. Transformations on the Coordinate Plane
Identify the coordinates of each vertex (plot the pre-image points).
Plug the coordinates into the rule.
Plot the Image points and connect.
Identify the transformations (reflection, rotation, dilation, translation).

14. Rules of Reflections Reflection x-axis y-axis origin y = x
Pre-image to image
(a, b)  (a, -b)
(a, b)  (-a, b)
(a, b)  (-a, -b)
(a, b)  (b, a)
Find coordinates
Multiply y coordinate by -1
Multiply x coordinate by -1
Multiply both coordinates by -1
Interchange x and y coordinates

15. Transformations on the Coordinate Plane
Algebra 4-2 Transformations on the Coordinate Plane
Transformations on the Coordinate Plane
Harbour

16. Reflections – continued…
Reflection across the x-axis: the x values stay the same and the y values change sign (x , y)  (x, -y)
Reflection across the y-axis: the y values stay the same and the x values change sign (x , y)  (-x, y)
Example:
In this figure, line l : n l
reflects across the y axis to line n
(2, 1)  (-2, 1) & (5, 4)  (-5, 4)
reflects across the x axis to line m.
(2, 1)  (2, -1) & (5, 4)  (5, -4) m
Lesson 10-5: Transformations

17. Reflects the figure over the y-axis
Use the given rule to transform the figure. Then describe the transformation.
Rule: (-x, y)
0pposite of x’s
Preimage
Image
(1,3)
(-1,3)
(1,1)
(-1,1)
Reflects the figure over the y-axis
(4,1)
(-4,1)

18. Reflects the figure over the x-axis
Use the given rule to transform the figure. Then describe the transformation.
Rule: (x, -y)
0pposite of y’s
Preimage
Image
(1,5)
(1,-5)
(1,3)
(1,-3)
Reflects the figure over the x-axis
(4,3)
(4,-3)

19. Reflections across specific lines:
To reflect a figure across the line y = a or x = a, mark the corresponding points equidistant from the line. i.e. If a point is 2 units above the line its corresponding image point must be 2 points below the line.
Example:
Reflect the fig. across the line y = 1.
(2, 3)  (2, -1).
(-3, 6)  (-3, -4)
(-6, 2)  (-6, 0)
Lesson 10-5: Transformations

20. Use the given rule to transform the figure
Use the given rule to transform the figure. Then describe the transformation.
Rule: (-x, -y)
opposite of x’s
opposite of y’s
Preimage
Image
(4,5)
(-4,-5)
(2,1)
(-2,-1)
Rotates the figure 180°
(4,1)
(-4,-1)

21. What letter would you get if you reflected each shape in its corresponding mirror line?

22. What letter would you get if you reflected each shape in its corresponding mirror line?

23. What letter would you get if you reflected each shape in its corresponding mirror line?

24. What letter would you get if you reflected each shape in its corresponding mirror line?

25. What letter would you get if you reflected each shape in its corresponding mirror line?

26. What letter would you get if you reflected each shape in its corresponding mirror line?

27. Reflection (x, y) (–x, y) W(–1, 4) (1, 4) X(4, 4) (–4, 4)
A trapezoid has vertices W(–1, 4), X(4, 4), Y(4, 1) and Z(–3, 1).
Trapezoid WXYZ is reflected over the y-axis. Find the coordinates of the vertices of the image.
To reflect the figure over the y-axis, multiply each x-coordinate by –1.
Answer: The coordinates of the vertices of the image are W(1, 4), X(–4, 4), Y(–4, 1), and Z(3, 1).
(x, y) (–x, y)
W(–1, 4) (1, 4)
X(4, 4) (–4, 4)
Y(4, 1) (–4, 1)
Z(–3, 1) (3, 1)
Example 2-2a

28. Reflection
A trapezoid has vertices W(–1, 4), X(4, 4), Y(4, 1), and Z(–3, 1).
Graph trapezoid WXYZ and its image W X Y Z.
Answer:
Graph each vertex of the trapezoid WXYZ. Connect the points. X W W X
Graph each vertex of the reflected image W X Y Z. Connect the points. Y Z Z Y
Example 2-2b

29. Reflection
A parallelogram has vertices A(–4, 7), B(2, 7), C(0, 4) and D(–2, 4).
a. Parallelogram ABCD is reflected over the x-axis. Find the coordinates of the vertices of the image.
Answer: A(–4, –7), B(2, –7), C(0, –4), D(–2, –4)
Example 2-2c

30. Reflection b. Graph parallelogram ABCD and its image A B C D.
Answer:
Example 2-2c

31. Warm-Up #35 Wednesday, 5/18
How far from the base of the house do you need to place a 15-foot ladder so that it is exactly reaches the top of a 12-foot tall wall?
Find the perimeter and area of polygon ABCD B C D A

32. Homework
Translations of Shapes worksheet

33. Translations of Shapes

34. Translations:
You “slide” a shape up, down, right, left or all the above. Translation is a rigid transformation. The new image will be congruent (isometry) to the Pre-image. Notation: (x, y) ( x + 2, y - 3)

35. Summary of Translations
Add to x
Translates RIGHT
Subtract from x
Translates LEFT
Add to y
Translates UP
Subtract from y
Translates DOWN

36. Transformations on the Coordinate Plane
Algebra 4-2 Transformations on the Coordinate Plane
Transformations on the Coordinate Plane
Harbour

37. Describe each transformation.
(x+10, y)
(x–5, y)
(x, y+7)
(x, y–6)
(x+3,y–7)
(x–4,y–5)
(x–8,y+9)
translates right 10
translates left 5
translates up 7
translates down 6
translates right 3
and down 7
translates left 4
and down 5
translates left 8
and up 9

38. translated figure up 3 units
Use the given rule to transform the figure. Then describe the transformation.
Rule: (x, y+3)
Add 3 to the y’s.
Preimage
Image
A (1,3)
A’(1, 6)
B (1,1)
B’(1, 4)
translated figure up 3 units
C (4,1)
C’(4, 4)

39. translated figure right 5 units
Use the given rule to transform the figure. Then describe the transformation.
Rule: (x+5, y)
Add 5 to the x’s.
Preimage
Image
A (-4,3)
A’ (1, 3)
B (-4,1)
B’ (1, 1)
translated figure right 5 units
C (-1,1)
C’ (4, 1)

40. translated figure left 2 units
Use the given rule to transform the figure. Then describe the transformation.
Rule: (x-2, y)
Subtract 2 from x’s
Preimage
Image
A (-3,-2)
A’ (-5, -2)
B (-3,-4)
B’ (-5, -4)
translated figure left 2 units
C (0,-4)
C’ (-2, -4)

41. translated figure down 4 units
Use the given rule to transform the figure. Then describe the transformation.
Rule: (x, y–4)
Subtract 4 from y’s.
Preimage
Image
A (2,1)
A’ (2, -3)
B (2,-1)
B’ (2, -5)
translated figure down 4 units
C (5,-1)
C’ (5, -5)

42. y A A’ B B’ C C’ x Image Transformation (x, y) (x + 5, y + 0)
Pre-image
A (-2, 4)
B (-3, 2)
C (-1, 1)
Image
A’ (3, 4)
B’ (2, 2)
C’ (4, 1)

43. y A A’ B C B’ C’ x Image Transformation (x, y) (x - 3, y + 0)
Pre-image
A (-2, 4)
B (-3, 2)
C (-1, 1)
Image
A’ (-5, 4)
B’ (-6, 2)
C’ (-4, 1)

44. y A B C x A’ B’ C’ Transformation (x, y) (x + 0, y - 5) Pre-image

45. y A’ B’ A C’ B C x Image Transformation (x, y) (x + 0, y + 4)
Pre-image
A (-2, 4)
B (-3, 2)
C (-1, 1)
Image
A’ (-2, 8)
B’ (-3, 6)
C’ (-1, 5)

46. y A B C A’ x B’ C’ Transformation (x, y) (x + 3, y - 4) Pre-image

47. y A’ A B’ B C’ C x Transformation (x, y) (x + 5, y + 2) Pre-image

48. y A B C x A’ B’ Image C’ Transformation (x, y) (x - 4, y - 5)
Pre-image
A (-2, 4)
B (-3, 2)
C (-1, 1)
Image
A’ (-6, -1)
B’ (-7, -3)
C’ (-5, -4) C’

49. y A’ B’ A C’ B C x Image Transformation (x, y) (x - 2, y + 3)
Pre-image
A (-2, 4)
B (-3, 2)
C (-1, 1)
Image
A’ (-4, 7)
B’ (-5, 5)
C’ (-3, 4)

50. Translation Triangle ABC has vertices A(–2, 1), B(2, 4), and C(1, 1).
Find the coordinates of the vertices of the image if it is translated 3 units to the right and 5 units down.
To translate the triangle 3 units to the right, add 3 to the x-coordinate of each vertex. To translate the triangle 5 units down, add –5 to the y-coordinate of each vertex.
Answer: The coordinates of the vertices of the image are A(1, –4), B(5, –1), and C(4, –4).
Example 2-3a

51. Translation Graph triangle ABC and its image. Answer: B
The preimage is . A
The translated image is C B A C
Example 2-3b

52. Translation Triangle JKL has vertices J(2, –3), K(4, 0), and L(6, –3).
a. Find the coordinates of the vertices of the image if it is translated 5 units to the left and 2 units up.
b. Graph triangle JKL and its image.
Answer: J(–3, –1), K(–1, 2), L(1, –1)
Answer:
Example 2-3c

53. Dilations of shapes

54. Dilations
A dilation is a type of transformation that enlarges or reduces a figure but the shape stays the same.
The dilation is described by a scale factor and a center of dilation.

55. Dilations
The scale factor k is the ratio of the length of any side in the image to the length of its corresponding side in the pre-image. It describes how much the figure is enlarged or reduced.

56. The dilation is a reduction if the scale factor is between 0 and 1.
Two types of dilations
Enlargement
Reduction
The dilation is a reduction if the scale factor is between 0 and 1.
The dilation is an enlargement if the scale factor is > 1.

57. Steps to Follow Plot the given points.
Multiply each coordinate by the scale factor.
Plot the image points.
State the coordinates of the dilation.

58. 4) Algebraic to verbal (2, 3) a) (x, y)  (2x, 2y) __________________ New coordinates: ( ) b) (x, y)  (1/4x, 1/4y)__________________ New coordinates: ( ) c) (x, y)  (2.5x, 2.5y)_____________________ d) (x, y)  (2y, 2x)_______________________

59. Given the vertices of the triangle, find a dilation by a scale factor of 3.x B’ C’
A (1,2)
B (3,3)
C (1,3)
A’ (3,6) A’
B’ (9,9) C B
C’ (3,9) A

60. Given the vertices of the rectangle, find a dilation by a scale factor of 2/3.x
A (-6,-3)
B (-6,3)
C (6,3)
D (6,-3)
A’ (-4,-2)
B’ (-4,2)
C’ (4,2) B C
D’ (4,-2) B’ C’ A’ D’ A D

61. Scale factor of 2 Scale factor of 1/2 Scale factor of 3
DILATION
SHAPE:
A (-2, -4)
B (3, -2)
C (1, 2)
D (-4,0)
Color Red
Scale factor of 2
Scale factor of 1/2
Scale factor of 3
Algebraic Representation
(x, y)  ( )
(x, y)  ( )
(x, y)  ( )
New Coordinates:
A’ ( )
B’ ( )
C’ ( )
D’ ( )
Color Blue
A’’ ( )
B’’ ( )
C’’ ( )
D”( )
Color Green
A’’’ ( )
B’’’ ( )
C’’’ ( )
D’’’ ( )
Color Pink

62. Reflections of Shapes

63. Transformations on the Coordinate Plane
Algebra 4-2 Transformations on the Coordinate Plane
Transformations on the Coordinate Plane
Harbour

64. Rotation Triangle ABC has vertices A(1, –3), B(3, 1), and C(5, –2).
Find the coordinates of the image of ABC after it is rotated 180° about the origin.
To find the coordinates of the image of ABC after a 180° rotation, multiply both coordinates of each point by –1.
Answer: The coordinates of the vertices of the image are A(–1, 3), B(–3, –1), and C(–5, 2).
Example 2-5a

65. Rotation Graph the preimage and its image. Answer: A C
The preimage is . B
The translated image is B C A
Example 2-5b

66. Rotation Triangle RST has vertices R(4, 0), S(2, –3), and T(6, –3).
a. Find the coordinates of the image of RST after it is rotated 90° counterclockwise about the origin.
b. Graph the preimage and the image.
Answer: R(0, 4), S(3, 2), T (3, 6)
Answer:
Example 2-5c

67. Transformations on the Coordinate Plane
Algebra 4-2 Transformations on the Coordinate Plane
Transformations on the Coordinate Plane
Harbour

68. Dilation
A trapezoid has vertices E(–1, 2), F(2, 1), G(2, –1), and H(–1, –2).
Find the coordinates of the dilated trapezoid E F G H if the scale factor is 2.
To dilate the figure, multiply the coordinates of each vertex by 2.
Answer: The coordinates of the vertices of the image are E(–2, 4), F(4, 2), G(4, –2), and H(–2, –4).
Example 2-4a

69. Dilation Graph the preimage and its image. Answer: E
The preimage is trapezoid EFGH. E F F
The image is trapezoid E F G H . G
Notice that the image has sides that are twice the length of the sides of the original figure. H G H
Example 2-4b

70. Dilation if the scale factor is
A trapezoid has vertices E(–4, 7), F(2, 7), G(0, 4), and H(–2, 4).
a. Find the coordinates of the dilated trapezoid E F G H
if the scale factor is
Answer:
Example 2-4c

71. Dilation
b. Graph the preimage and its image.
Answer:
Example 2-4c

72. Example 1
Identify the transformation. Then use arrow notation to describe the transformation.
A. B. C.

73. Example 2
Name the transformation.
ABC A’B’C’
A’B’C’ A”B”C”

74. B C D A K N M L

75. Example 4
Show that the transformation is an isometry by using the Distance Formula to compare the side lengths of the triangles.

76. Find the value of each variable given that the transformation
Example 5
Find the value of each variable given that the transformation
is an isometry.
3x – 8 7
3y°
45°
2z + 13 25