what shape will result from taking a cross section of the rectangular prism through the points shown?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Cross Sections of 3D Shapes" ], "new_concepts": [], "current_concepts": [ "Cross Sections of 3D Shapes" ] } ## Identify the locations of the three given points The three orange points lie on the edges of the rectangular prism: - One point is on the front-left vertical edge. - One point is on the top-left-back edge (receding edge). - One point is on the top-front horizontal edge. ## Determine the plane passing through the three points A unique plane is defined by these three non-collinear points. To find the cross section, we trace the intersection of this plane with the faces of the rectangular prism: - The segment connecting the point on the front-left vertical edge and the point on the top-left-back edge lies entirely on the left face of the prism. - The segment connecting the point on the front-left vertical edge and the point on the top-front horizontal edge lies entirely on the front face of the prism. - The segment connecting the point on the top-left-back edge and the point on the top-front horizontal edge lies entirely on the top face of the prism. ## Classify the resulting cross-sectional shape The intersection of the plane with the three faces (left, front, and top) forms a closed three-sided polygon. Therefore, the resulting cross section is a triangle. What shape will result from taking a cross section of the rectangular prism through the points shown? triangle { "elements": [ { "type": "point", "params": [[-3, -2]], "properties": { "name": "A", "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "size": 4, "withLabel": false } }, { "type": "point", "params": [[3, -2]], "properties": { "name": "B", "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "size": 4, "withLabel": false } }, { "type": "point", "params": [[3, 2]], "properties": { "name": "C", "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "size": 4, "withLabel": false } }, { "type": "point", "params": [[-3, 2]], "properties": { "name": "D", "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "size": 4, "withLabel": false } }, { "type": "point", "params": [[-1, 4]], "properties": { "name": "E", "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "size": 4, "withLabel": false } }, { "type": "point", "params": [[5, 4]], "properties": { "name": "F", "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "size": 4, "withLabel": false } }, { "type": "point", "params": [[5, 0]], "properties": { "name": "G", "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "size": 4, "withLabel": false } }, { "type": "point", "params": [[-1, 0]], "properties": { "name": "H", "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "size": 4, "withLabel": false } }, { "type": "segment", "params": [[-3, -2], [3, -2]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[3, -2], [3, 2]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[3, 2], [-3, 2]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[-3, 2], [-3, -2]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[3, 2], [5, 4]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[-3, 2], [-1, 4]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[-1, 4], [5, 4]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[5, 4], [5, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[5, 0], [3, -2]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[-1, 0], [-1, 4]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 1, "dash": 2 } }, { "type": "segment", "params": [[-1, 0], [5, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 1, "dash": 2 } }, { "type": "segment", "params": [[-1, 0], [-3, -2]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 1, "dash": 2 } }, { "type": "point", "params": [[-3, 0]], "properties": { "name": "P1", "color": "#F28A55", "strokeColor": "#F28A55", "fillColor": "#F28A55", "size": 5, "withLabel": false } }, { "type": "point", "params": [[-2, 3]], "properties": { "name": "P2", "color": "#F28A55", "strokeColor": "#F28A55", "fillColor": "#F28A55", "size": 5, "withLabel": false } }, { "type": "point", "params": [[0, 2]], "properties": { "name": "P3", "color": "#F28A55", "strokeColor": "#F28A55", "fillColor": "#F28A55", "size": 5, "withLabel": false } }, { "type": "polygon", "params": [[-3, 0], [-2, 3], [0, 2]], "properties": { "fillColor": "rgba(242, 138, 85, 0.4)", "strokeColor": "#F2557F", "strokeWidth": 2 } } ], "timestamps": [0.1] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Cross Sections of 3D Shapes" ] }

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Explanation:

Identify the locations of the three given points

Determine the plane passing through the three points

Classify the resulting cross-sectional shape

Answer: