draw the image of $\triangle abc$ under a dilation whose center is $p$ and scale factor is 3.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Geometric Transformations", "Center of Dilation", "Dilation Scale Factor" ], "new_concepts": [], "current_concepts": [ "Geometric Transformations", "Center of Dilation", "Dilation Scale Factor", "Coordinate Dilation" ] } ## Identify coordinates of the center of dilation and vertices $$ \begin{aligned} &P = (0, 0) \ &A = (-1, 2) \ &B = (-2, -1) \ &C = (3, 0) \end{aligned} $$ ## Apply dilation formula with scale factor 3 $$ \begin{aligned} &k = 3 \ &A' = (3 \cdot (-1), 3 \cdot 2) = (-3, 6) \ &B' = (3 \cdot (-2), 3 \cdot (-1)) = (-6, -3) \ &C' = (3 \cdot 3, 3 \cdot 0) = (9, 0) \end{aligned} $$ ## Verify the dilated triangle vertices $$ \begin{aligned} &A' = (-3, 6) \ &B' = (-6, -3) \ &C' = (9, 0) \end{aligned} $$ The vertices of the image $\triangle A'B'C'$ under a dilation centered at $P$ with a scale factor of $3$ are: $$ \begin{aligned} A' &= (-3, 6) \ B' &= (-6, -3) \ C' &= (9, 0) \end{aligned} $$ { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "P", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[-1, 2]], "properties": { "name": "A", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[-2, -1]], "properties": { "name": "B", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[3, 0]], "properties": { "name": "C", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "polygon", "params": [[-1, 2], [-2, -1], [3, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "fillColor": "rgba(85, 131, 242, 0.15)" } }, { "type": "point", "params": [[-3, 6]], "properties": { "name": "A'", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[-6, -3]], "properties": { "name": "B'", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[9, 0]], "properties": { "name": "C'", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "polygon", "params": [[-3, 6], [-6, -3], [9, 0]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "fillColor": "rgba(140, 85, 242, 0.2)" } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Dilation Scale Factor" ] }

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Explanation:

Identify coordinates of the center of dilation and vertices

Apply dilation formula with scale factor 3

Verify the dilated triangle vertices

Answer: