warm-up
what is the image of $\delta abc$ after a dilation by a scale factor of 3 centered at the origin?
use the drawing tool(s) to form the correct answer on the provided grid.

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Dilation in Coordinate Plane" ], "new_concepts": [ "Dilation of Polygons" ], "current_concepts": [ "Dilation in Coordinate Plane", "Dilation of Polygons" ] } ## Identify original coordinates Using the *Dilation in Coordinate Plane* knowledge point $$ \begin{aligned} &A = (1, 1) \ &B = (2, 0) \ &C = (0, 0) \end{aligned} $$ ## Apply scale factor Using the *Dilation in Coordinate Plane* knowledge point $$ \begin{aligned} &(x, y) \to (3x, 3y) \ &A' = (3 \cdot 1, 3 \cdot 1) = (3, 3) \ &B' = (3 \cdot 2, 3 \cdot 0) = (6, 0) \ &C' = (3 \cdot 0, 3 \cdot 0) = (0, 0) \end{aligned} $$ The vertices of the image $\Delta A'B'C'$ after a dilation by a scale factor of 3 centered at the origin are: $$ \begin{aligned} A' &= (3, 3) \ B' &= (6, 0) \ C' &= (0, 0) \end{aligned} $$ To form the correct answer on the grid, plot the points $(3, 3)$, $(6, 0)$, and $(0, 0)$, and connect them with line segments to form the dilated triangle. { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "C", "size": 3, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[1, 1]], "properties": { "name": "A", "size": 3, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[2, 0]], "properties": { "name": "B", "size": 3, "color": "#583C87", "withLabel": true } }, { "type": "polygon", "params": [[0, 0], [1, 1], [2, 0]], "properties": { "strokeColor": "#583C87", "strokeWidth": 1, "fillColor": "rgba(88, 60, 135, 0.1)" } }, { "type": "point", "params": [[0, 0]], "properties": { "name": "C'", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[3, 3]], "properties": { "name": "A'", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[6, 0]], "properties": { "name": "B'", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "polygon", "params": [[0, 0], [3, 3], [6, 0]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "fillColor": "rgba(140, 85, 242, 0.3)" } } ], "timestamps": [0.5, 1.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Dilation in Coordinate Plane" ] }

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Question

Explanation:

Identify original coordinates

Apply scale factor

Answer: