find the composition of transformations that map abcd to abcd.

rotate clockwise about the origin ? °, then reflect over the -axis.

Question

find the composition of transformations that map abcd to abcd.

rotate clockwise about the origin  ? °, then reflect over the  -axis.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Coordinate Rotation", "Rotation Rules", "Coordinate Reflection", "Composite Transformations" ], "new_concepts": [], "current_concepts": [ "Coordinate Rotation", "Rotation Rules", "Coordinate Reflection", "Composite Transformations" ] } ## Identify coordinates of original and final vertices Using the **Coordinate Translation** knowledge point $$ \begin{aligned} &A(-6, 6), \quad B(-4, 6), \quad C(-4, 2), \quad D(-6, 2)\ &A'(6, -6), \quad B'(6, -4), \quad C'(2, -4), \quad D'(2, -6) \end{aligned} $$ ## Apply clockwise rotation about the origin Using the **Coordinate Rotation** and **Rotation Rules** knowledge points $$ \begin{aligned} &\text{Let } R_{90^\circ\text{ CW}}(x, y) = (y, -x)\ &A(-6, 6) \to A''(6, 6)\ &B(-4, 6) \to B''(6, 4)\ &C(-4, 2) \to C''(2, 4)\ &D(-6, 2) \to D''(2, 6) \end{aligned} $$ ## Apply reflection to match final coordinates Using the **Coordinate Reflection** and **Composite Transformations** knowledge points $$ \begin{aligned} &\text{Compare } A''(6, 6) \text{ with } A'(6, -6)\ &\text{Reflect over the } x\text{-axis: } (x, y) \to (x, -y)\ &A''(6, 6) \to (6, -6) = A'\ &B''(6, 4) \to (6, -4) = B'\ &C''(2, 4) \to (2, -4) = C'\ &D''(2, 6) \to (2, -6) = D' \end{aligned} $$ Find the composition of transformations that map ABCD to A'B'C'D'. Rotate clockwise about the origin 90$^\circ$, then reflect over the x-axis. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Composite Transformations" ] }

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Question

Explanation:

Identify coordinates of original and final vertices

Apply clockwise rotation about the origin

Apply reflection to match final coordinates

Answer: