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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Dilation in Coordinate Plane" ], "current_concepts": [ "Dilation in Coordinate Plane", "Geometric Transformations" ] } ## Identify the vertices of the original polygon $ABCD$ $$ A(1, 2), \quad B(3, 4), \quad C(6, 4), \quad D(6, 2) $$ ## Analyze the first target image $$ \begin{aligned} &A'(2, 2), \quad B'(3, 3), \quad C'(4.5, 3), \quad D'(4.5, 2)\ &\text{Scale factor } k = 0.5 \text{ centered at } (3, 2):\ &(x', y') = (3 + 0.5(x - 3), 2 + 0.5(y - 2))\ &A': (3 + 0.5(1 - 3), 2 + 0.5(2 - 2)) = (2, 2)\ &B': (3 + 0.5(3 - 3), 2 + 0.5(4 - 2)) = (3, 3)\ &C': (3 + 0.5(6 - 3), 2 + 0.5(4 - 2)) = (4.5, 3)\ &D': (3 + 0.5(6 - 3), 2 + 0.5(2 - 2)) = (4.5, 2) \end{aligned} $$ ## Analyze the second target image $$ \begin{aligned} &A'(-2, 1), \quad B'(1, 4), \quad C'(5.5, 4), \quad D'(5.5, 1)\ &\text{Scale factor } k = 1.5 \text{ centered at } (4, 4):\ &(x', y') = (4 + 1.5(x - 4), 4 + 1.5(y - 4))\ &A': (4 + 1.5(1 - 4), 4 + 1.5(2 - 4)) = (-0.5, 1) \quad (\text{not matching})\ &\text{Scale factor } k = 1.5 \text{ centered at } (7, 4):\ &(x', y') = (7 + 1.5(x - 7), 4 + 1.5(y - 4))\ &A': (7 + 1.5(1 - 7), 4 + 1.5(2 - 4)) = (-2, 1)\ &B': (7 + 1.5(3 - 7), 4 + 1.5(4 - 4)) = (1, 4)\ &C': (7 + 1.5(6 - 7), 4 + 1.5(4 - 4)) = (5.5, 4)\ &D': (7 + 1.5(6 - 7), 4 + 1.5(2 - 4)) = (5.5, 1) \end{aligned} $$ ## Analyze the third target image $$ \begin{aligned} &A'(-1, 0), \quad B'(3, 4), \quad C'(9, 4), \quad D'(9, 0)\ &\text{Scale factor } k = 2 \text{ centered at } (3, 4):\ &(x', y') = (3 + 2(x - 3), 4 + 2(y - 4))\ &A': (3 + 2(1 - 3), 4 + 2(2 - 4)) = (-1, 0)\ &B': (3 + 2(3 - 3), 4 + 2(4 - 4)) = (3, 4)\ &C': (3 + 2(6 - 3), 4 + 2(4 - 4)) = (9, 4)\ &D': (3 + 2(6 - 3), 4 + 2(2 - 4)) = (9, 0) \end{aligned} $$ The correct matches for the dilations of polygon $ABCD$ to their corresponding images are: * **First Image** (smallest scaled-down figure): Dilation centered at $(3, 2)$ with a scale factor of $0.5$ * **Second Image** (medium scaled-up figure shifted left): Dilation centered at $(7, 4)$ with a scale factor of $1.5$ * **Third Image** (largest scaled-up figure sharing vertex $B$): Dilation centered at $(3, 4)$ with a scale factor of $2$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Dilation in Coordinate Plane" ] }

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Identify the vertices of the original polygon \(ABCD\)

Analyze the first target image

Analyze the second target image

Analyze the third target image

Answer: