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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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.

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)

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)