draw the image of $\triangle abc$ under a dilation whose center is $a$ and scale factor is $\frac{1}{4}$.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Geometric Transformations", "Dilation Scale Factor", "Center of Dilation" ], "new_concepts": [], "current_concepts": [ "Geometric Transformations", "Dilation Scale Factor", "Center of Dilation" ] } ## Identify the coordinates of the vertices $$ \begin{aligned} A &= (-7, 1) \ B &= (5, 5) \ C &= (1, -7) \end{aligned} $$ ## Apply the dilation formula with center A and scale factor k = 1/4 $$ \begin{aligned} P' &= A + k(P - A) \ A' &= A = (-7, 1) \ B' &= (-7, 1) + \frac{1}{4}((5, 5) - (-7, 1)) = (-7, 1) + \frac{1}{4}(12, 4) = (-4, 2) \ C' &= (-7, 1) + \frac{1}{4}((1, -7) - (-7, 1)) = (-7, 1) + \frac{1}{4}(8, -8) = (-5, -1) \end{aligned} $$ ## Verify the coordinates of the image triangle $$ \begin{aligned} A' &= (-7, 1) \ B' &= (-4, 2) \ C' &= (-5, -1) \end{aligned} $$ The vertices of the image triangle $\Delta A'B'C'$ under the dilation centered at $A$ with a scale factor of $\frac{1}{4}$ are: $$ \begin{aligned} A' &= (-7, 1) \ B' &= (-4, 2) \ C' &= (-5, -1) \end{aligned} $$ { "elements": [ { "type": "point", "params": [[-7, 1]], "properties": { "name": "A (A')", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[5, 5]], "properties": { "name": "B", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[1, -7]], "properties": { "name": "C", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "polygon", "params": [[-7, 1], [5, 5], [1, -7]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "fillColor": "rgba(140, 85, 242, 0.1)" } }, { "type": "point", "params": [[-4, 2]], "properties": { "name": "B'", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[-5, -1]], "properties": { "name": "C'", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "polygon", "params": [[-7, 1], [-4, 2], [-5, -1]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 3, "fillColor": "rgba(242, 85, 127, 0.3)" } } ], "timestamps": [0.5, 1.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Dilation Scale Factor" ] }

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Explanation:

Identify the coordinates of the vertices

Apply the dilation formula with center A and scale factor k = 1/4

Verify the coordinates of the image triangle

Answer: