1. WARM-UP 8 in. Perimeter = _____ 7 in. Area = _____ 12 in. 4 cm
Circumference = _____
Area = _____
Perimeter = _____
5 in
Area = _____
17 ft
25 mm
14 ft

2. SOLUTIONS Perimeter = _____ 30 in. 7 in. Area = _____ 56 sq. in.
4 cm
SOLUTIONS
Circumference = _____
25.12 cm
Area = _____
50.24 sq. cm
Perimeter = 40_
40 mm
119 sq. ft
Area = _____
17 ft
25 mm
14 ft

3. Find the Perimeter and area
A = ½ (b1+b2)h
A = ½ bh
A =L w
A=Лr2 ÷ 2
A = (58) (40)
A = ½(40+76 )58
A = ½(58) (36)
A = (3.14) (20)2
A = 2,320 units2
A = 3364 units2
A =1,256 ÷ 2 units2
A = 1044 units2
A =628 units2
A = 3364 units2
+ 628 units2
=3,992 units2

4. Congruent Figures
Two figures have the same shape & same size)
X and Y are congruent. X Y
If two figures are congruent, then they will fit exactly on each other.
Find out by inspection the congruent figures among the following. A B C D E F G H
B, D ;
C, F

5. TRANSFORMATIONS

6. Transformations Image (transformed figure)
Changes position or orientation of a figure.
(preserves size & shape but changes location)
Each point of original figure is paired with exactly one point of its image on the plane.
Image (transformed figure)
(congruent to original figure.)
We can identify a symmetry as a transformation of the plane that moves the pattern so that it falls back on itself. The only transformations that we'll consider are those that preserve distance, called isometries. (Self-similar fractals have symmetries on different scales, and so other transformations must be considered to understand them.) There are four kinds of planar isometries: translations, rotations, reflections, and glide reflections
indicated with “prime” notation

7. 3 Common Transformations
1. Translation, (to slide )
2. Reflection, (flip or mirror image )
Read slide…
There are other transformations … really combinations of these three…
click when completed...
3. Rotation, (a turn a side around a point.)

8. Reflection

9. REFLECTION divides figure into congruent parts FLIP Line of symmetry-
(splits image down middle.
Creates a mirror image.)

10. LINE OF SYMMETRY
Figure on the right shows a symmetric figure with l being the axis of symmetry. Are there are any congruent figures.
The line l divides the figure into 2 congruent figures, i.e and are congruent figures.
Thus, 2 congruent figures. X

11. Creates a mirror image.)
REFLECTION
FLIP
Line of symmetry-
Creates a mirror image.)

12. Reflection (flip) specified by a line of reflection, like a mirror.
Can be:
vertical
Horizontal
diagonal
A reflection fixes one line in the plane, called the axis of reflection, and exchanges points on one side of the axis with points on the other side of the axis at the same distance from the axis. In the example you see a diagonal axis in white. The double-ended red arrows are supposed to suggest the reflection. It's called a "reflection" because similar things happen with a reflection in a mirror.
REFLECTION: When you look in a mirror, your image is reflected back at you. If you imagine that your image actually exists on the other side of the mirror, you get some idea of the mathematical definition of reflection or mirror image. We can reflect figures by folding along the mirror line as you see below.
specified by a line of reflection, like a mirror.

13. “crawl before you climb”
Reflect over the y axis
(10,3)
(2,4)
(0,1)
“crawl before you climb”

14. “crawl before you climb”
Reflect over the x axis
(-10,-3)
(-2,-4)
(0,-1)
“crawl before you climb”

15. “crawl before you climb”
Reflect over the y axis
(-10,-3)
(-2,-4)
(0,-1)
“crawl before you climb”

16. Symmetry of the Alphabet
Asymmetric =(not mirror image of graph along axis)
Symmetric = (mirror image of graph along axis)
Symmetry of the Alphabet
Sort capital letters of alphabet into groups according to symmetries
Divide letters into two categories:
symmetrical
not symmetrical
Symmetrical: A, B, C, D, E, H, I, K, M, N, O, S, T, U, V, W, X, Y, Z
Not Symmetrical: F, G, J, L, P, Q, R

17. Rotation

18. Rotation (turn) A rotation is a turn through an angle. specified by:
center of rotation
an angle measure.
rotational symmetry
a turn of n degrees,
around a center point,
A rotation fixes one point in the plane and turns the rest of it some angle around that point. In general a rotation could be by any angle, but for patterns like we have, the angle has to divide 360°, and a little more analysis finds further restrictions. In fact, the angle of rotation can only be 180°, 120°, 90°, or 60°. The order of a rotation is the number of times it has to be performed to bring the plane back to where it started. So a 60°-rotation has order 6, a 90°-rotation has order 4, a 120°-rotation has order 3, and an 180°-rotation has order 2. A 180° rotation is also called a half turn.
ROTATION: To rotate is to turn about a point. When you make a left-hand turn at a corner, we say you are rotating 90 degrees about the corner.

19. rotates a figure around a point.
Rotation
To rotate must have:
a center
an angle
rotates a figure around a point.

20. Rotation is simply turning about a fixed point.
Here the fixed point is the origin
Rotate
90 counterclockwise
about the origin
Rotate
180
about the origin
Rotate 90clockwise about the origin

21. Rotate 90 degrees clockwise.
(like a right turn)
Hands in the air on the wheel.
Left hand: x
Right hand: y
Make a clockwise turn.
Which hand is at 12 o’clock 1st? X
Rotate 90 degrees clockwise.
1. Change the sign of x
2. Switch order of x & y.

22. Rotate 90 degrees clockwise.

23. Rotate 90° clockwise

24. Rotate 90° clockwise

25. Rotate 90 degrees counterclockwise.
is a left turn.
Hands in the air on the wheel.
Left hand: x
Right hand: y
Make a counterclockwise turn.
Which hand is at 12 o’clock 1st? Y
Rotate 90 degrees counterclockwise.
1. Change sign of y
2. Switch the order of x and y

26. Example: Rotate 90 degrees counterclockwise.

27. Rotate 90° counterclockwise

28. Rotate 90° counterclockwise

29. Rotating 180 degrees changes the sign of the x and the sign of the y.
Rotate 180 degrees.
Rotating 180 degrees changes the sign of the x and the sign of the y.
Keep the order
&
change the sign of both x & y.

30. Example: Rotate 180 degrees.

31. Rotate 180°

32. Rotate 180°

33. Homework Does the Brain Good.

34. Translation (slide) TRANSLATION a slide without turning.
If you have a given repeating pattern, you can slide it along a certain direction a certain distance and it will fall back upon itself with all the patterns exactly matching. This symmetry is called a translation.
TRANSLATION: Translate means to slide. A translation moves a figure a given distance in a given direction. You can think of a translation as sliding an image across a piece of paper.
specified by a direction and a distance.

35. Translation (SLIDE) ‘Prime notation’
Each point of R is “moved” to a new position R’
R’ is the image of R.
‘Prime notation’
image
Translation (SLIDE) l
Move without rotating or reflecting
Every translation has a direction and a distance
moves a figure
over,
down,
or up.

37. Use‘Transformation notation’ to move an image.
(x,y)→(x+3, y)
(x,y)
(x,y)
(-6,-2)→(-6+3, -2)→(-3, -2)
(x,y)
(-6,-5)→(-6+3, -5)→(-3, -5)
(x,y)
(-4,-5)→(-4+3, -5)→(-1, -5)

41. Can you do same transformation twice?
Or combine more than 2 transformations?
(x,y)→(x+5, y)

42. You Try (Create the new image by translating them.
(x,y)→(x+5, y)
(x,y)→(x-1, y + 4)

43. (x,y)→(x+5, y) (-6,-2)→(-6+5, -2)→(-1, -2) (x,y) (x,y)
(-6,-5)→(-6+5, -5)→(-1, -5)
(x,y)
(-4,-5)→(-4+5, -5)→ (1, -5)
(x,y) l

44. (x,y)→(x-1, y + 4)
(-6,-2)→(-6-1, )→(-7, 2)
(x,y)
(x,y)
(-6,-5)→(-6-1, )→(-7, -1)
(x,y)
(-4,-5)→(- 4-1, )→ (-5, -1)
(x,y)

45. Red is obtained by transforming blue one about point x.
which type of transformation (translation, rotation, reflection, enlargement, reduction) & are they congruent?
(a)
Reflection
Yes
(b)
Translation
Yes
(c)
____________
Rotation
Yes

46. Left 3 and up 1 Down 5 (-3, 5), (-4, 0), (0, -3), (1, 2)
TRY THESE
1. Explain how a figure is translated for (-3, 1).
Explain how a figure is translated for (0, -5).
Reflect over the y-axis: (3, 5), (4, 0), (0, -3), (-1, 2).
Reflect over the x-axis: (-5, 2), (3, 0), (-1, -2).
Left 3 and up 1
Down 5
(-3, 5), (-4, 0), (0, -3), (1, 2)
(-5, -2), (3, 0), (-1, 2)

47. SYMMETRY

48. 2 types of symmetry:
1. Reflectional Symmetry
2. Rotational Symmetry

49. Symmetry Reflection Rotational Symmetry (rotates around a set point)
Look at the figure at right and does it have:
Symmetry?
Rotational symmetry? If so what angles?
Rotate the figure around the center
Reflection
Rotational Symmetry
(rotates around a set point)

51. A figure is turned about a point and it coincides with the original, it has rotational symmetry.
How many degrees has this figure rotated?
120 degrees
How?
360/3 = 120

52. point symmetry
(figure with rotational symmetry of 180o)

53. A rotation of 360 degrees is called a
rotational identity.

54. Every figure has symmetry of rotational identity.
Therefore, every figure
must have
at least one symmetry.

55. Determine if each figure has rotational symmetry.
If so, list the degree of turn needed to complete the rotation.
Point Symmetry
90 degrees
Rotational Identity
60 degrees
90 degrees

56. Reflectional Symmetry
is also called
Line Symmetry.
Horizontal
Vertical
Vertical and Horizontal

57. GLIDE
REFLECTION
transformation (movement of one figure where each point of original figure is paired with exactly one point of its image on the plane.
Isometry (transformation that preserves size & shape but changes location)
Image (always congruent to original figure.)

58. Glide Reflection Involves more than one translation
Combines reflection and translation
Required:
Reflection line must be parallel to direction of slide.

59. Glide reflection
glide reflection (special product of a reflection & a translation along line of reflection.)
only type of symmetry involving more than one step.
The fourth kind of isometry, the glide reflection, is not nearly as easy to see as the other three. It's composed of a reflection across an axis and a translation along the axis.
GLIDE REFLECTION: This transformation combines translations and reflections. A glide reflection occurs when you slide an image in one direction and then reflect it over a line.

60. What type is each transformation?
All points slide same distance in same direction.
Translation
picture is “turned”.
Rotation
image is “flipped” across a line.
Reflection
2-step process: slide & flip.
Glide reflection

61. Group Activity Choose a letter (other than R) with no symmetries
On a piece of paper perform the following tasks on the chosen letter:
rotation
translation
reflection
glide reflection

62. Transformations Four Types: TO Slide, flip, or turn a figure/pattern.
Reflection
Rotation
Translation
Glide Reflection
Isometry (transformation that preserves size & shape but changes location)

64. Transformation and Congruence B)
11.1 The Meaning of Congruence
1B_Ch11(64)
Example
Transformation and Congruence B)
‧ When a figure is translated, rotated or reflected, the image produced is congruent to the original figure. When a figure is enlarged or reduced, the image produced will NOT be congruent to the original one.
Index 11.1
Index

65. (c)
Enlargement No
(e)
____________
Reduction No

66. SYMMETRY Activity Take out a piece of paper.
Pour 5 drops of paint in the center of paper.
Fold the paper in half.
Open the paper and let it dry.

67. Reflection (flip or mirror reflection)
A reflection is determined by a line in the plane called the line of reflection.
Each point P of the plane is transformed to the point P’ on the opposite side of the line of reflection and the same distance from the line of reflection.

68. 1. The image of point (3,-5) under the translation that shifts (x,y) to (x-1,y-3) is.....
A. (-4,8) B. (2,8) C. (-3,15) D. (2,-8)
image of point (3,-5) under translation that shifts (x,y) to (x-1,y-3) is (2,-8)
2. A translation maps (x,y) (x+1,y+2) what are
the coordinates of B (-2,4) after translation?
coordinates of B(-2,4) after translation: (-1,6).
3. What is the image of point P(-3,2) under the transformation T(2,6)?
T(-2,6) means add -2 to the x-value (-3) & +6 to the y-value (2). image of point P is (-5,8).

69. What is the image of point P(4,2) under the transformation T(-2,2)?
The image of point P(4,2) under the transformation T(-2,2) is (2,4).
What is the image of point P(-2,-7) under the transformation T(6,4)?
The image of point P(-2,-7) under the transformation T(6,4) is (4,-3).
If the point (4,1) has a translation of (-2,4), what are the coordinates of pt. (-1,5) under the same translation?
To solve this problem, you have to add (-2,4) to the point (-1,5). Always add x-value with x-value and vice-versa (For example: add -2 to -1 and add 4 to 5.) The answer is (-3,9).

70. If the coordinates of the vertices of triangle ABC are A(-4,-1), B(-1,5) and C(2,1), what are the coordinates of triangle A'B'C', the translation of triangle ABC, under T(4,3)?
The coordinates of triangle A‘ B‘ C' are A'(0,2), B'(3,8) and C'(6,4).
How do we translate points & figures in a coordinate plane?