1. Math 8
Day 11
Learning Target: Students can identify the image of a figure after a combined transformation is performed, and
determine whether the final image is similar or congruent to the original.

2. Transformation Sequences
Many famous geometric designs are created by combining transformations of various geometric figures.
The mysterious crop circles shown here appear to be composed of combined rotations and dilations. If you consider the one piece that is rotated to create the design, you can see that it is similar to each of its
rotations.

3. Transformation Sequences
Example 1:
Identify the combined transformations from the original to the final image, and tell whether the two figures are similar or congruent.
Original: A(1, -1), B(2, –1), C(3, –5), D(2, –5)
First Image: A'(1, 1), B'(2, 1), C'(3, 5), D'(2, 5)
Final Image: A"(–3, –2) B"(–2, –2) C"(–1, 2) D"(–2, 2)
The original image to the first image: Reflection across x-axis
The first image to the final image: Translation 4 units left and 3 units down
The two figures are congruent.

4. Transformation Sequences
Example 2:
Identify the combined transformations from the original to the final image, and tell whether the two figures are similar or congruent.
Original: A(2, –1), B(4, –1), C(5, –5), D(3, –5)
First Image: A'(2, 1), B'(4, 1), C'(5, 5), D'(3, 5)
Final Image: A"(–5, –5), B"(–3, –5), C"(–2, –1), D"(–4, –1)
The original image to the first image: Reflection across x-axis
The first image to the final image: Translation 7 units left and 6 units down
The two figures are congruent.

5. Transformation Sequences
Example 3:
Identify the combined transformations from the original to the final image, and tell whether the two figures are similar or congruent.
Original: A(1, 0), B(3, 1), C(4, –1), D(2, –2)
First Image: A'(–1, 0), B'(–3, –1), C'(–4, 1), D'(–2, 2)
Final Image: A"(–0.5, 0) B"(–1.5, –0.5) C"(–2, 0.5) D"(–1, 1)
The original image to the first image: Rotation of 180 around origin
The first image to the final image: dilation with scale factor of 0.5
The two figures are similar.

6. Transformation Sequences
Example 4:
Identify the combined transformations from the original to the final image, and tell whether the two figures are similar or congruent.
The original image to the first image: Translation3 units left and 2 units down
The first image to the final image: dilation with scale factor of 2
The two figures are similar.

7. Additional Example 3: Finding Sequences of Transformations
Transformation Sequences
Additional Example 3: Finding Sequences of Transformations
Find a sequence of at least two combined transformations from the original to the final image. Justify your answer.
Original : A(–5, 1), B(–2, 4), C(–1, 1)
Final Image: A"(1, –5), B"(4, –2), C"(1, –1)
Reflection across the x-axis; rotation of 90 counter-clockwise around the origin.

8. Transformation Sequences
Check It Out: Example 3
Find a sequence of at least two combined transformations from the original to the final image. Justify your answer.
Original : A(–5, 1), B(–2, 4), C(–1, 1)
Final Image: A"(–5, –1), B"(–2, –4), C"(–1, –1)
Reflection across the x-axis; translation 1 unit up

9. Lesson Quiz for Student Response Systems
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems 9 9

10. First image : Δ A'B'C': A' (2, -1), B'(4, -3), C '(2, -3)
Lesson Quiz : Part-I
Identify the combined transformations from the original to the final image, and tell whether the two figures are similar or congruent.
1. Original Δ ABC: A(1, 2), B(3, 4),
C(3, 2)
First image : Δ A'B'C': A' (2, -1),
B'(4, -3), C '(2, -3)
Final image Δ A"B"C " :
A"(-1, 3), B"(1, 1), C "(-1, 1)
A 90° clockwise rotation around the origin followed by a translation of 3 units left and 4 units up; congruent

11. 2. Original ABCD: A(-2, 1), B(1, 1), C(0, 2), D(-2, 2)
Lesson Quiz : Part-II
2. Original ABCD: A(-2, 1), B(1, 1),
C(0, 2), D(-2, 2)
First image A'B'C 'D': A'(-2, -1),
B'(1, -1), C '(0, –2), D'(-2, -2)
Final image A"B"C "D": A"(-4, -2),
B"(2, -2), C "(0, -4), D"(-4, -4)
A reflection across the x-axis followed by a dilation with a scale factor of 2 with the origin as the center of dilation; similar

12. 180° rotation around the origin followed by a translation of 1 unit
Lesson Quiz : Part-III
Find a sequence of at least two combined transformations from the original to the final image.
180° rotation around the origin followed by a translation of 1 unit
right and 1 unit down.

13. Lesson Quiz for Student Response System
1. Triangle ABC has the following coordinates:
A(–5, 1), B(–2, 4), C(–1, 1). Which triangle coordinates represent a reflection across the y-axis?
A. A’(–5, 1), B’(–2, 4), C’(–1, 1)
B. A’(5, 1), B’(2, 4), C’(1, 1)
C. A’(–5, –1), B’(–2, –4), C’(–1, –1)
D. A’(1, –5), B’(4, –2), C’(1, –1)

14. Lesson Quiz for Student Response System
2. Triangle ABC has the following coordinates: A(–5, 1), B(–2, 4), C(–1, 1). Which triangle coordinates represent a 90 clockwise rotation around the origin?
A. A’(–5, 1), B’(–2, 4), C’(–1, 1)
B. A’(5, 1), B’(2, 4), C’(1, 1)
C. A’(–5, –1), B’(–2, –4), C’(–1, –1)
D. A’(1, 5), B’(4, 2), C’(1, 1)