identify which shapes on the graph are congruent to shape i by performing these sequences of transformations on shape i:

* a reflection across the $y$-axis, followed by a $90^\circ$ counterclockwise rotation about the origin, and then a translation 3 units down
* a $90^\circ$ counterclockwise rotation about the origin and then a translation 2 units up and 2 units left
* a $180^\circ$ rotation about the origin and then a translation 1 unit right

Question

identify which shapes on the graph are congruent to shape i by performing these sequences of transformations on shape i:

* a reflection across the $y$-axis, followed by a $90^\circ$ counterclockwise rotation about the origin, and then a translation 3 units down
* a $90^\circ$ counterclockwise rotation about the origin and then a translation 2 units up and 2 units left
* a $180^\circ$ rotation about the origin and then a translation 1 unit right

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Geometric Transformations", "Congruence Transformations" ], "new_concepts": [], "current_concepts": [ "Geometric Transformations", "Congruence Transformations" ] } ## Identify the coordinates of the reference point on Shape I $$ \begin{aligned} &\text{Shape I is the small blue shape at the bottom, centered near } (0, -1).\ &\text{Let the reference center point of Shape I be } P = (0, -1). \end{aligned} $$ ## Apply the first sequence of transformations $$ \begin{aligned} &\text{Sequence 1: Reflection across } y\text{-axis, } 90^\circ\text{ CCW rotation about origin, translation } 3\text{ units down.}\ &P(0, -1) \xrightarrow{\text{Reflect } y} (0, -1) \xrightarrow{\text{Rotate } 90^\circ\text{ CCW}} (1, 0) \xrightarrow{\text{Translate } 3\text{ down}} (1, -3) \quad (\text{No shape here}) \end{aligned} $$ $$ \begin{aligned} &\text{Let's re-verify with a vertex of Shape I: } V = (0, -1) \text{ (top vertex), } B = (0, -1.5) \text{ (bottom vertex).}\ &\text{Since Shape I is a small blue triangle/polygon at } (0, -1), \text{ let's test the other sequences.} \end{aligned} $$ ## Apply the second sequence of transformations $$ \begin{aligned} &\text{Sequence 2: } 90^\circ\text{ CCW rotation about origin, translation } 2\text{ units up and } 2\text{ units left.}\ &P(0, -1) \xrightarrow{\text{Rotate } 90^\circ\text{ CCW}} (1, 0) \xrightarrow{\text{Translate } 2\text{ up, } 2\text{ left}} (-1, 2) \quad (\text{No shape here}) \end{aligned} $$ ## Apply the third sequence of transformations $$ \begin{aligned} &\text{Sequence 3: } 180^\circ\text{ rotation about origin, translation } 1\text{ unit right.}\ &P(0, -1) \xrightarrow{\text{Rotate } 180^\circ} (0, 1) \xrightarrow{\text{Translate } 1\text{ right}} (1, 1)\ &\text{The green star shape is centered at } (1.5, 1) \text{ or } (1, 1). \text{ Let's check the vertices of Shape I:}\ &\text{Shape I has vertices at } (0, -1), (-0.5, -1), (0.5, -1), (0, -1.3).\ &\text{After } 180^\circ\text{ rotation: } (0, 1), (0.5, 1), (-0.5, 1), (0, 1.3).\ &\text{After translating } 1\text{ unit right: } (1, 1), (1.5, 1), (0.5, 1), (1, 1.3).\ &\text{This matches the small blue/green triangular base at the bottom of the green star shape at } y = 1. \end{aligned} $$ The shape on the graph that is congruent to Shape I by performing the third sequence of transformations (a $180^\circ$ rotation about the origin and then a translation 1 unit right) is the **green star shape** (specifically its bottom triangular component centered at $(1, 1)$). { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Geometric Transformations" ] }

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Question

Explanation:

Identify the coordinates of the reference point on Shape I

Apply the first sequence of transformations

Apply the second sequence of transformations

Apply the third sequence of transformations

Answer: