1. Drill: Monday, 9/28
Find the coordinates of the image of ∆ABC with vertices A(3, 4), B(–1, 4), and C(5, –2), after each reflection.
1. across the x-axis
A’(3, –4), B’(–1, –4), C’(5, 2)
2. across the y-axis
A’(–3, 4), B’(1, 4), C’(–5, –2)
3. across the line y = x
A’(4, 3), B’(4, –1), C’(–2, 5)
OBJ: SWBAT identify and draw translations.

2. Unit B Extending Transformational Geometry

3. 12-1
Reflections
Holt McDougal Geometry
Holt Geometry

5. Example 4A: Drawing Reflections in the Coordinate Plane
Reflect the figure with the given vertices across the given line.
X(2, –1), Y(–4, –3), Z(3, 2); x-axis
The reflection of (x, y) is (x,–y). Y’
X(2,–1) X’(2, 1) Z X’
Y(–4,–3) Y’(–4, 3) X
Z(3, 2) Z’(3, –2) Z’ Y
Graph the image and preimage.

6. Example 4B: Drawing Reflections in the Coordinate Plane
Reflect the figure with the given vertices across the given line.
R(–2, 2), S(5, 0), T(3, –1); y = x S’ R’ T’
The reflection of (x, y) is (y, x).
R(–2, 2) R’(2, –2) S R T
S(5, 0) S’(0, 5)
T(3, –1) T’(–1, 3)
Graph the image and preimage.

7. Reflect the rectangle with vertices S(3, 4),
Check It Out! Example 4
Reflect the rectangle with vertices S(3, 4),
T(3, 1), U(–2, 1) and V(–2, 4) across the x-axis.
The reflection of (x, y) is (x,–y).
S(3, 4) S’(3, –4) V S U T
T(3, 1) T’(3, –1)
U(–2, 1) U’(–2, –1) V’ S’ U’ T’
V(–2, 4) V’(–2, –4)
Graph the image and preimage.

8. An isometry is a transformation that does not change the shape or size of a figure. Reflections, translations, and rotations are all isometries. Isometries are also called congruence transformations or rigid motions.

9. B2 Translations
Copy: B2 CN, B2 Practice B,
GT Geometry

11. Example 1: Identifying Translations
Tell whether each transformation appears to be a translation. Explain. A. B.
No; the figure appears to be flipped.
Yes; the figure appears to slide.

12. Check It Out! Example 1
Tell whether each transformation appears to be a translation. a. b.
Yes; all of the points
have moved the same
distance in the same
direction.
No; not all of the points have moved the same distance.

15. Example 3: Drawing Translations in the Coordinate Plane
Translate the triangle with vertices D(–3, –1), E(5, –3), and F(–2, –2) along the vector <3, –1>.
The image of (x, y) is (x + 3, y – 1).
D(–3, –1) D’(–3 + 3, –1 – 1) = D’(0, –2)
E(5, –3) E’(5 + 3, –3 – 1) = E’(8, –4)
F(–2, –2) F’(–2 + 3, –2 – 1) = F’(1, –3)
Graph the preimage and the image.

16. The image of (x, y) is (x – 3, y – 3).
Check It Out! Example 3
Translate the quadrilateral with vertices R(2, 5), S(0, 2), T(1,–1), and U(3, 1) along the vector <–3, –3>.
The image of (x, y) is (x – 3, y – 3). R S T U R’ S’ T’ U’
R(2, 5) R’(2 – 3, 5 – 3) = R’(–1, 2)
S(0, 2) S’(0 – 3, 2 – 3) = S’(–3, –1)
T(1, –1) T’(1 – 3, –1 – 3) = T’(–2, –4)
U(3, 1) U’(3 – 3, 1 – 3) = U’(0, –2)
Graph the preimage and the image.

17. Example 1: Identifying Rotations
Tell whether each transformation appears to be a rotation. Explain. B. A.
No; the figure appears
to be flipped.
Yes; the figure appears
to be turned around a point.

18. Check It Out! Example 1
Tell whether each transformation appears to be a rotation. b. a.
Yes, the figure appears to be turned around a point.
No, the figure appears to be a translation.

19. If the angle of a rotation in the coordinate plane is not a multiple of 90°, you can use sine and cosine ratios to find the coordinates of the image.

20. Example 3: Drawing Rotations in the Coordinate Plane
Rotate ΔJKL with vertices J(2, 2), K(4, –5), and L(–1, 6) by 180° about the origin.
The rotation of (x, y) is (–x, –y).
J(2, 2) J’(–2, –2)
K(4, –5) K’(–4, 5)
L(–1, 6) L’(1, –6)
Graph the preimage and image.

21. Check It Out! Example 3
Rotate ∆ABC by 180° about the origin.
The rotation of (x, y) is (–x, –y).
A(2, –1) A’(–2, 1)
B(4, 1) B’(–4, –1)
C(3, 3) C’(–3, –3)
Graph the preimage and image.

22. Lesson Quiz: Part II
Rotate ∆RST with vertices R(–1, 4), S(2, 1), and T(3, –3) about the origin by the given angle.
3. 90°
R’(–4, –1), S’(–1, 2), T’(3, 3)
4. 180°
R’(1, –4), S’(–2, –1), T’(–3, 3)

23. Examples
Determine the coordinates of the image of P(4, –7) under each transformation.
1. a translation 3 units left and 1 unit up
(1, –6)
2. a rotation of 90° about the origin
(7, 4)
3. a reflection across the y-axis
Copy: B4 CN, B4 CW, Transf. Board Game
(–4, –7)
OBJ: SWBAT identify and draw compositions of transformations.

24. B4 Compositions
Honors Geometry

25. A composition of transformations is one transformation followed by another.
For example, a glide reflection is the composition of a translation and a reflection across a line parallel to the translation vector.

26. The glide reflection that maps ∆JKL to ∆J’K’L’ is the composition of a translation along followed by a reflection across line l.

27. The image after each transformation is congruent to the previous image
The image after each transformation is congruent to the previous image. By the Transitive Property of Congruence, the final image is congruent to the preimage. This leads to the following theorem.

28. Example 1B: Drawing Compositions of Isometries
Draw the result of the composition of isometries. K L M
∆KLM has vertices K(4, –1), L(5, –2), and M(1, –4). Rotate ∆KLM 180° about the origin and then reflect it across the y-axis.

29. Step 1 The rotational image of (x, y) is (–x, –y).
Example 1B Continued
Step 1 The rotational image of (x, y) is (–x, –y). M’ K’ L’ L” M” K”
K(4, –1)  K’(–4, 1),
L(5, –2)  L’(–5, 2), and
M(1, –4)  M’(–1, 4).
Step 2 The reflection image of (x, y) is (–x, y). K L M
K’(–4, 1)  K”(4, 1),
L’(–5, 2)  L”(5, 2), and M’(–1, 4)  M”(1, 4).
Step 3 Graph the image and preimages.

30. Check It Out! Example 1
∆JKL has vertices J(1,–2), K(4, –2), and L(3, 0). Reflect ∆JKL across the x-axis and then rotate it 180° about the origin. L K J

31. Check It Out! Example 1 Continued
Step 1 The reflection image of (x, y) is (–x, y).
J(1, –2) J’(–1, –2), K(4, –2) K’(–4, –2), and L(3, 0) L’(–3, 0). J” K” L'
Step 2 The rotational image of (x, y) is (–x, –y).
L'’ K’ J’ L K J
J’(–1, –2) J”(1, 2), K’(–4, –2) K”(4, 2), and L’(–3, 0) L”(3, 0).
Step 3 Graph the image and preimages.

33. Lesson Quiz: Part I
PQR has vertices P(5, –2), Q(1, –4), and P(–3, 3).
1. Translate ∆PQR along the vector <–2, 1> and then reflect it across the x-axis.
P”(3, 1), Q”(–1, –5), R”(–5, –4)
2. Reflect ∆PQR across the line y = x and then rotate it 90° about the origin.
P”(–5, –2), Q”(–1, 4), R”(3, 3)

34. Dilations

35. Definitions
Scale Factor: The scale factor is the number we multiply the sides or coordinates of a figure by to get the new coordinates or sides.
Enlargement: When you multiply by a number whose absolute value is greater then 1. (Gets Bigger)
Reduction: When you multiply by a number whose absolute value is less than 1. (Gets Smaller)

36. Identifying Dilations
Previously, you studied rigid transformations, in which the image and preimage of a figure are congruent. In this lesson, you will study a type of nonrigid transformation called a dilation, in which the image and preimage of a figure are similar.
A dilation with center C and scale factor k is a transformation that maps every point P in the plane to a point P so that the following properties are true.
If P is not the center point C, then the image point P lies on
The scale factor k is a positive number such that k = , and k  1. CP
CP 
If P is the center point C, then P = P .

37. • • Identifying Dilations
The dilation is a reduction if 0 < k < 1 and it is an enlargement if k > 1. P P´ 6 5 P´ P 3 2 Q´ Q • • R C C R´ Q R´ Q´ R
Reduction: k = = = 3 6 1 2
CP CP
Enlargement: k = = 5 2
CP CP
Because PQR ~ P´Q´R´, is equal to the scale factor of the dilation.
P´Q´ PQ

38. • Dilation in a Coordinate Plane
In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky).
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4).
Use the origin as the center and use a scale factor of How does the perimeter of the preimage compare to the perimeter of the image? 1 2
SOLUTION y
Because the center of the dilation is the origin, you can find the image of each vertex by multiplying its coordinates by the scale factor. D C
A(2, 2)  A´(1, 1) D´ A C´ B A´
B(6, 2)  B´(3, 1) 1 B´ •
C(6, 4)  C ´(3, 2) O 1 x
D(2, 4)  D´(1, 2)

39. • Dilation in a Coordinate Plane
In a coordinate plane, dilations whose centers are the origin have the property that the image of P(x, y) is P´(kx, ky).
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4). Use the origin as the center and use a scale factor of How does the perimeter of the preimage compare to the perimeter of the image? 1 2
SOLUTION y
From the graph, you can see that the preimage has a perimeter of 12 and the image has a perimeter of 6. D C D´ A C´ B
A preimage and its image after a dilation are similar figures. A´ 1 B´ •
Therefore, the ratio of the perimeters of a preimage and its image is equal to the scale factor of the dilation. O 1 x

40. Example 1
Identify the dilation and find its scale factor C

41. Example 1 Identify the dilation and find its scale factor
Reduction, Because
The scale factor is k = ⅔
Enlargement, Because
The scale factor is k = 2

42. Example 2 Dilation in a Coordinate Plane
Draw a dilation of a rectangle ABCD with vertices A(1,1), B(3,1), C(3,2) and D(1,2). Use the origin as the center and use a scale factor of 2. How does the perimeter of the preimage compare to the perimeter of the image?

43. Dilation in a Coordinate Plane
Draw a dilation of a rectangle ABCD with vertices A(1,1), B(3,1), C(3,2) and D(1,2). Use the origin as the center and use a scale factor of 2. How does the perimeter of the preimage compare to the perimeter of the image?
Preimage has a perimeter of 6
Image has a perimeter of 12
The perimeter was enlarged by a factor of 2

44. Symmetry Objective: Today, we will identify types of
symmetry in figures.

45. Reflectional Symmetry/Line Symmetry
A figure has reflectional symmetry if and only if a line coincides with the original figure.
The line is called the axis of symmetry.

46. Reflectional Line of Symmetry
A figure has reflectional symmetry if and only if there exists a line that “cuts” the figure into two congruent parts, that fall on top of each other when folded over the line of symmetry.

47. Rotational Symmetry
A figure has rotational symmetry of “n” degrees if you can rotate the figure “n” degrees and get the exact same image.
N must be between 0 and 360.

48. Rotational Symmetry
Wind Mill
How many degrees can each rotate?

49. Point Symmetry
A figure has point symmetry when a rotation of 180 degrees maps the figure onto itself.
(It looks exactly the same upside down)

50. Examples
Name all the types of symmetry each figure has: (if rotational state how many degrees)
Rotational Symmetry (180)
Or Point Symmetry
Reflectional Symmetry (1)
Reflectional Symmetry (8)
Rotational Symmetry (45)

51. Which category does it fit?