L’espace - 4e
Repérage dans un pavé droit
Exercice 1 : Lire les coordonnées d'un point dans un plan 3D
Dans le repère ci-dessous, donner les coordonnées du point M sous la forme \((x ; y ; z)\).
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0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_80": {"type": "Segment", "pt1": {"x": 54.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 54.0, "y": 0.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_81": {"type": "Point", "position": {"x": 0.0, "y": 0.0, "z": 54.0}, "point_render": {"visible": false}}, "Label_for_Point3D_81": {"type": "Label", "position": {"x": 0.0, "y": 0.0, "z": 54.0}, "text": "9", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}}, "Segment3D_82": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 54.0}, "pt2": {"x": 0.0, "y": 1.2, "z": 54.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_83": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 54.0}, "pt2": {"x": 1.2, "y": 0.0, "z": 54.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_84": {"type": "Point", "position": {"x": 0.0, "y": 60.0, "z": 0.0}, "point_render": {"visible": false}}, "Label_for_Point3D_84": {"type": "Label", "position": {"x": 0.0, "y": 60.0, "z": 0.0}, "text": "10", "position_offset": {"x": 3.52, "y": 0.0, "z": 0.0}}, "Segment3D_85": {"type": "Segment", "pt1": {"x": 0.0, "y": 60.0, "z": 0.0}, "pt2": {"x": 1.2, "y": 60.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_86": {"type": "Segment", "pt1": {"x": 0.0, "y": 60.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 60.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_87": {"type": "Point", "position": {"x": 60.0, "y": 0.0, "z": 0.0}, "point_render": {"visible": false}}, "Label_for_Point3D_87": {"type": "Label", "position": {"x": 60.0, "y": 0.0, "z": 0.0}, "text": "10", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}}, "Segment3D_88": {"type": "Segment", "pt1": {"x": 60.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 60.0, "y": 1.2, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_89": {"type": "Segment", "pt1": {"x": 60.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 60.0, "y": 0.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_90": {"type": "Point", "position": {"x": 0.0, "y": 0.0, "z": 60.0}, "point_render": {"visible": false}}, "Label_for_Point3D_90": {"type": "Label", "position": {"x": 0.0, "y": 0.0, "z": 60.0}, "text": "10", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}}, "Segment3D_91": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 60.0}, "pt2": {"x": 0.0, "y": 1.2, "z": 60.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_92": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 60.0}, "pt2": {"x": 1.2, "y": 0.0, "z": 60.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_93": {"type": "Point", "position": {"x": 0.0, "y": 0.0, "z": 0.0}, "point_render": {"visible": false}}, "Label_for_Point3D_93": {"type": "Label", "position": {"x": 0.0, "y": 0.0, "z": 0.0}, "text": "0", "position_offset": {"x": 0.0, "y": -3.52, "z": 0.0}}, "Segment3D_94": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 1.2, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_95": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 1.2, "y": 0.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_96": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 0.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_97": {"type": "Point", "position": {"x": 42.0, "y": 54.0, "z": 54.0}, "point_render": {"color": 204}, "label_render": {"color": 204}}, "Label_for_Point3D_97": {"type": "Label", "position": {"x": 42.0, "y": 54.0, "z": 54.0}, "text": "M", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}, "label_render": {"color": 204}}, "Point3D_98": {"type": "Point", "position": {"x": 42.0, "y": 0.0, "z": 54.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Point3D_99": {"type": "Point", "position": {"x": 42.0, "y": 54.0, "z": 0.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Point3D_100": {"type": "Point", "position": {"x": 0.0, "y": 54.0, "z": 54.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Segment3D_101": {"type": "Segment", "pt1": {"x": 42.0, "y": 54.0, "z": 54.0}, "pt2": {"x": 42.0, "y": 0.0, "z": 54.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}, "Segment3D_102": {"type": "Segment", "pt1": {"x": 42.0, "y": 54.0, "z": 54.0}, "pt2": {"x": 42.0, "y": 54.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}, "Segment3D_103": {"type": "Segment", "pt1": {"x": 42.0, "y": 54.0, "z": 54.0}, "pt2": {"x": 0.0, "y": 54.0, "z": 54.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}}}
Exercice 2 : Se repérer dans un pavé droit
On représente le pavé droit \( ABCDEFGH \) dans le repère d'origine \( A \) ci-dessous.
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Donner les coordonnées du point \( A \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
Donner les coordonnées du point \( H \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
Sachant que \( L ∈ [AB] \), donner les coordonnées du point \( L \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
Placer le point de coordonnées \( ( 3 ; 0 ; 3 ) \)
Exercice 3 : Indiquer si un entier n (entre 101 et 1000) est divisible par 2, 3, 5, 9, 10
95 est-il divisible par les nombres suivants ? Cocher les bonnes réponses.
580 est-il divisible par les nombres suivants ? Cocher les bonnes réponses.
1960 est-il divisible par les nombres suivants ? Cocher les bonnes réponses.
375 est-il divisible par les nombres suivants ? Cocher les bonnes réponses.
Exercice 4 : Lire les coordonnées d'un point dans un plan 3D
Dans le repère ci-dessous, donner les coordonnées du point M sous la forme \((x ; y ; z)\).
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"Label_for_Point3D_1": {"type": "Label", "position": {"x": 72.0, "y": 0.0, "z": 0.0}, "text": "y", "position_offset": {"x": 0.0, "y": -3.52, "z": 0.0}}, "Point3D_2": {"type": "Point", "position": {"x": 0.0, "y": 72.0, "z": 0.0}, "point_render": {"visible": false}}, "Label_for_Point3D_2": {"type": "Label", "position": {"x": 0.0, "y": 72.0, "z": 0.0}, "text": "z", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}}, "Point3D_3": {"type": "Point", "position": {"x": 0.0, "y": 6.0, "z": 0.0}, "point_render": {"visible": false}}, "Label_for_Point3D_3": {"type": "Label", "position": {"x": 0.0, "y": 6.0, "z": 0.0}, "text": "1", "position_offset": {"x": 3.52, "y": 0.0, "z": 0.0}}, "Segment3D_4": {"type": "Segment", "pt1": {"x": 0.0, "y": 6.0, "z": 0.0}, "pt2": {"x": 1.2, "y": 6.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_5": {"type": "Segment", "pt1": {"x": 0.0, "y": 6.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 6.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_6": {"type": "Point", "position": {"x": 6.0, "y": 0.0, "z": 0.0}, "point_render": {"visible": false}}, "Label_for_Point3D_6": {"type": "Label", "position": {"x": 6.0, "y": 0.0, "z": 0.0}, "text": "1", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}}, "Segment3D_7": {"type": "Segment", "pt1": {"x": 6.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 6.0, "y": 1.2, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_8": {"type": "Segment", "pt1": {"x": 6.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 6.0, "y": 0.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_9": {"type": "Point", "position": {"x": 0.0, "y": 0.0, "z": 6.0}, "point_render": {"visible": false}}, "Label_for_Point3D_9": {"type": "Label", "position": {"x": 0.0, "y": 0.0, "z": 6.0}, "text": "1", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}}, "Segment3D_10": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 6.0}, "pt2": {"x": 0.0, "y": 1.2, "z": 6.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_11": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 6.0}, "pt2": {"x": 1.2, "y": 0.0, "z": 6.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_12": {"type": "Point", "position": {"x": 0.0, "y": 12.0, "z": 0.0}, "point_render": {"visible": false}}, "Label_for_Point3D_12": {"type": "Label", "position": {"x": 0.0, "y": 12.0, "z": 0.0}, "text": "2", "position_offset": {"x": 3.52, "y": 0.0, "z": 0.0}}, "Segment3D_13": {"type": "Segment", "pt1": {"x": 0.0, "y": 12.0, "z": 0.0}, "pt2": {"x": 1.2, "y": 12.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_14": {"type": "Segment", "pt1": {"x": 0.0, "y": 12.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 12.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_15": {"type": 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"z": 0.0}, "label_render": {"color": 204}}, "Point3D_98": {"type": "Point", "position": {"x": 36.0, "y": 0.0, "z": 12.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Point3D_99": {"type": "Point", "position": {"x": 36.0, "y": 42.0, "z": 0.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Point3D_100": {"type": "Point", "position": {"x": 0.0, "y": 42.0, "z": 12.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Segment3D_101": {"type": "Segment", "pt1": {"x": 36.0, "y": 42.0, "z": 12.0}, "pt2": {"x": 36.0, "y": 0.0, "z": 12.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}, "Segment3D_102": {"type": "Segment", "pt1": {"x": 36.0, "y": 42.0, "z": 12.0}, "pt2": {"x": 36.0, "y": 42.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}, "Segment3D_103": {"type": "Segment", "pt1": {"x": 36.0, "y": 42.0, "z": 12.0}, "pt2": {"x": 0.0, "y": 42.0, "z": 12.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}}}
Exercice 5 : Se repérer dans un pavé droit
On représente le pavé droit \( ABCDEFGH \) dans le repère d'origine \( A \) ci-dessous.
{"init": {"range": [[-2.5, 5], [-5, 5]], "scale": [50, 50], "hasGraph": true, "axisArrows": "-", "axisOpacity": 0.5, "gridOpacity": 0.1, "gridStep": [0.5, 0.5], "tickStep": [10, 10], "yaxis": true}, "label": [[[0, 0], "A", "above left"], [[-1.5, -3], "B", "below left"], [[1.5, -3], "C", "below right"], [[3, 0], "D", "above right"], [[0, 4], "E", "above left"], [[-1.5, 1], "F", "above left"], [[1.5, 1], "G", "above right"], [[3, 4], "H", "above right"], [[-0.5, -3], "L", "above left"], [[-2, -4.5], "x", "below right", {"color": "#8BC63E"}], [[4.5, 0], "y", "below right", {"color": "#8BC63E"}], [[0, 4], "z", "above right", {"color": "#8BC63E"}], [[0, 0], "0", "below between", {"color": "#000000"}, {"opacity": "0.5"}], [[-0.5, -1], "1", "below between", {"color": "#000000"}, {"opacity": "0.5"}], [[1, 0], "1", "below between", {"color": "#000000"}, {"opacity": "0.5"}], [[0, 1], "1", "above right", {"color": "#000000"}, {"opacity": "0.5"}]], "line": [[[-2.5, -5], [-2.25, -5], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-2.5, -5], [-2.5, -4.75], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-3, -6], [0, 0], {"stroke": "#000000"}, {"opacity": "0.5"}], [[4.75, 0.25], [5, 0], {"stroke": "#000000"}, {"opacity": "0.5"}], [[5, 0], [4.75, -0.25], {"stroke": "#000000"}, {"opacity": "0.5"}], [[5, 0], [0, 0], {"stroke": "#000000"}, {"opacity": "0.5"}], [[0.25, 4.75], [0, 5], {"stroke": "#000000"}, {"opacity": "0.5"}], [[0, 5], [-0.25, 4.75], {"stroke": "#000000"}, {"opacity": "0.5"}], [[0, 5], [0, 0], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-0.4, -1.05], [-0.6, -0.95], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-0.9, -2.05], [-1.1, -1.95], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-1.4, -3.05], [-1.6, -2.95], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-1.9, -4.05], [-2.1, -3.95], {"stroke": "#000000"}, {"opacity": "0.5"}], [[1, -0.1], [1, 0.1], {"stroke": "#000000"}, {"opacity": "0.5"}], [[2, -0.1], [2, 0.1], {"stroke": "#000000"}, {"opacity": "0.5"}], [[3, -0.1], [3, 0.1], {"stroke": "#000000"}, {"opacity": "0.5"}], [[4, -0.1], [4, 0.1], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-0.1, 1], [0.1, 1], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-0.1, 2], [0.1, 2], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-0.1, 3], [0.1, 3], {"stroke": "#000000"}, {"opacity": "0.5"}], [[-0.1, 4], [0.1, 4], {"stroke": "#000000"}, {"opacity": "0.5"}], [[0, 0], [-1.5, -3], {"stroke": "#000000"}], [[0, 0], [3, 0], {"stroke": "#000000"}], [[-1.5, -3], [1.5, -3], {"stroke": "#000000"}], [[3, 0], [1.5, -3], {"stroke": "#000000"}], [[0, 4], [3, 4], {"stroke": "#000000"}], [[3, 4], [1.5, 1], {"stroke": "#000000"}], [[0, 4], [-1.5, 1], {"stroke": "#000000"}], [[-1.5, 1], [1.5, 1], {"stroke": "#000000"}], [[3, 4], [3, 0], {"stroke": "#000000"}], [[1.5, 1], [1.5, -3], {"stroke": "#000000"}], [[-1.5, 1], [-1.5, -3], {"stroke": "#000000"}], [[0, 4], [0, 0], {"stroke": "#000000"}], [[-0.1, 0], [0.1, 0], {"stroke": "#6495ED"}], [[0, -0.1], [0, 0.1], {"stroke": "#6495ED"}], [[-1.6, -3], [-1.4, -3], {"stroke": "#6495ED"}], [[-1.5, -3.1], [-1.5, -2.9], {"stroke": "#6495ED"}], [[1.4, -3], [1.6, -3], {"stroke": "#6495ED"}], [[1.5, -3.1], [1.5, -2.9], {"stroke": "#6495ED"}], [[2.9, 0], [3.1, 0], {"stroke": "#6495ED"}], [[3, -0.1], [3, 0.1], {"stroke": "#6495ED"}], [[-0.1, 4], [0.1, 4], {"stroke": "#6495ED"}], [[0, 3.9], [0, 4.1], {"stroke": "#6495ED"}], [[-1.6, 1], [-1.4, 1], {"stroke": "#6495ED"}], [[-1.5, 0.9], [-1.5, 1.1], {"stroke": "#6495ED"}], [[1.4, 1], [1.6, 1], {"stroke": "#6495ED"}], [[1.5, 0.9], [1.5, 1.1], {"stroke": "#6495ED"}], [[2.9, 4], [3.1, 4], {"stroke": "#6495ED"}], [[3, 3.9], [3, 4.1], {"stroke": "#6495ED"}], [[-0.6, -3], [-0.4, -3], {"stroke": "#6495ED"}], [[-0.5, -3.1], [-0.5, -2.9], {"stroke": "#6495ED"}]]}
Donner les coordonnées du point \( A \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
Donner les coordonnées du point \( H \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
Sachant que \( L ∈ [BC] \), donner les coordonnées du point \( L \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
On donnera la réponse sous la forme \( (x;y;z) \), par exemple \( (2;2;1) \).
Placer le point de coordonnées \( ( 0 ; 0 ; 3 ) \)