13) the vertices of a quadrilateral are (-3, 3), (4, 5), (6, -1), and (-1, -3). determine if the quadrilateral is a parallelogram.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Parallelogram Verification", "Slope Formula" ], "current_concepts": [ "Parallelogram Verification", "Slope Formula", "Coordinate Geometry" ] } ## Calculate slopes of opposite sides $$ \begin{aligned} &A(-3, 3), \quad B(4, 5), \quad C(6, -1), \quad D(-1, -3)\ &m_{AB} = \frac{5 - 3}{4 - (-3)} = \frac{2}{7}\ &m_{CD} = \frac{-3 - (-1)}{-1 - 6} = \frac{-2}{-7} = \frac{2}{7}\ &m_{BC} = \frac{-1 - 5}{6 - 4} = \frac{-6}{2} = -3\ &m_{DA} = \frac{3 - (-3)}{-3 - (-1)} = \frac{6}{-2} = -3 \end{aligned} $$ ## Compare slopes of opposite sides $$ \begin{aligned} &m_{AB} = m_{CD} = \frac{2}{7} \implies AB \parallel CD\ &m_{BC} = m_{DA} = -3 \implies BC \parallel DA \end{aligned} $$ ## Determine quadrilateral type $$ \text{Both pairs of opposite sides are parallel } \implies ABCD \text{ is a parallelogram.} $$ Yes, the quadrilateral is a parallelogram because both pairs of opposite sides are parallel (the slopes of opposite sides are equal: $m_{AB} = m_{CD} = \frac{2}{7}$ and $m_{BC} = m_{DA} = -3$). { "elements": [ { "type": "point", "params": [[-3, 3]], "properties": { "name": "A(-3,3)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[4, 5]], "properties": { "name": "B(4,5)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[6, -1]], "properties": { "name": "C(6,-1)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[-1, -3]], "properties": { "name": "D(-1,-3)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "polygon", "params": [[-3, 3], [4, 5], [6, -1], [-1, -3]], "properties": { "fillColor": "rgba(140, 85, 242, 0.2)", "strokeColor": "#8C55F2", "strokeWidth": 2 } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Parallelogram Verification" ] }

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Calculate slopes of opposite sides

Compare slopes of opposite sides

Determine quadrilateral type

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