Solides - 6e

Pavé droit, cube et prisme droit

Exercice 1 : Reconnaître le patron d'un pavé droit

Soit le pavé suivant :

Parmi les figures suivantes, trouver celle ou celles étant un patron de ce pavé.

A.

{"init": {"range": [[-1.25, 24.0], [-1.0, 31.25]], "scale": [20, 20]}, "label": [[[0.0, 4.0], "8 cm", "left", {}], [[11.5, 30.0], "9 cm", "above", {}], [[19.5, 23.0], "7 cm", "above", {}], [[7.0, 26.5], "7 cm", "left", {}], [[3.5, 8.0], "7 cm", "above", {}], [[7.0, 19.0], "8 cm", "left", {}], [[7.0, 11.5], "7 cm", "left", {}]], "line": [[[7.0, 0.0], [7.0, 8.0], {"subtype": "segment"}], [[7.0, 0.0], [16.0, 0.0], {"subtype": "segment"}], [[16.0, 15.0], [16.0, 23.0], {"subtype": "segment"}], [[7.0, 15.0], [7.0, 23.0], {"subtype": "segment"}], [[7.0, 30.0], [16.0, 30.0], {"subtype": "segment"}], [[7.0, 23.0], [16.0, 23.0], {"subtype": "segment"}], [[16.0, 23.0], [16.0, 30.0], {"subtype": "segment"}], [[0.0, 0.0], [7.0, 0.0], {"subtype": "segment"}], [[0.0, 8.0], [7.0, 8.0], {"subtype": "segment"}], [[23.0, 15.0], [23.0, 23.0], {"subtype": "segment"}], [[7.0, 8.0], [7.0, 15.0], {"subtype": "segment"}], [[7.0, 23.0], [7.0, 30.0], {"subtype": "segment"}], [[7.0, 8.0], [16.0, 8.0], {"subtype": "segment"}], [[16.0, 8.0], [16.0, 15.0], {"subtype": "segment"}], [[16.0, 23.0], [23.0, 23.0], {"subtype": "segment"}], [[7.0, 15.0], [16.0, 15.0], {"subtype": "segment"}], [[16.0, 0.0], [16.0, 8.0], {"subtype": "segment"}], [[16.0, 15.0], [23.0, 15.0], {"subtype": "segment"}], [[0.0, 0.0], [0.0, 8.0], {"subtype": "segment"}]]}
B.

{"init": {"range": [[-1.25, 33.0], [-1.0, 26.25]], "scale": [20, 20]}, "line": [[[0.0, 16.0], [0.0, 25.0], {"subtype": "segment"}], [[15.0, 0.0], [15.0, 8.0], {"subtype": "segment"}], [[15.0, 8.0], [24.0, 8.0], {"subtype": "segment"}], [[15.0, 16.0], [15.0, 25.0], {"subtype": "segment"}], [[32.0, 8.0], [32.0, 16.0], {"subtype": "segment"}], [[7.0, 16.0], [15.0, 16.0], {"subtype": "segment"}], [[15.0, 8.0], [15.0, 16.0], {"subtype": "segment"}], [[7.0, 25.0], [15.0, 25.0], {"subtype": "segment"}], [[0.0, 16.0], [7.0, 16.0], {"subtype": "segment"}], [[24.0, 8.0], [32.0, 8.0], {"subtype": "segment"}], [[24.0, 16.0], [32.0, 16.0], {"subtype": "segment"}], [[7.0, 16.0], [7.0, 25.0], {"subtype": "segment"}], [[15.0, 0.0], [24.0, 0.0], {"subtype": "segment"}], [[15.0, 16.0], [24.0, 16.0], {"subtype": "segment"}], [[24.0, 8.0], [24.0, 16.0], {"subtype": "segment"}], [[7.0, 8.0], [7.0, 16.0], {"subtype": "segment"}], [[7.0, 8.0], [15.0, 8.0], {"subtype": "segment"}], [[24.0, 0.0], [24.0, 8.0], {"subtype": "segment"}], [[0.0, 25.0], [7.0, 25.0], {"subtype": "segment"}]], "label": [[[15.0, 4.0], "8 cm", "left", {}], [[0.0, 20.5], "9 cm", "left", {}], [[19.5, 16.0], "9 cm", "above", {}], [[28.0, 16.0], "8 cm", "above", {}], [[11.0, 25.0], "8 cm", "above", {}], [[7.0, 12.0], "8 cm", "left", {}], [[3.5, 25.0], "7 cm", "above", {}]]}
C.

{"init": {"range": [[-1.25, 23.0], [-1.0, 33.25]], "scale": [20, 20]}, "line": [[[0.0, 0.0], [0.0, 9.0], {"subtype": "segment"}], [[22.0, 16.0], [22.0, 25.0], {"subtype": "segment"}], [[15.0, 16.0], [15.0, 25.0], {"subtype": "segment"}], [[15.0, 25.0], [15.0, 32.0], {"subtype": "segment"}], [[7.0, 0.0], [15.0, 0.0], {"subtype": "segment"}], [[7.0, 16.0], [15.0, 16.0], {"subtype": "segment"}], [[7.0, 25.0], [15.0, 25.0], {"subtype": "segment"}], [[7.0, 32.0], [15.0, 32.0], {"subtype": "segment"}], [[0.0, 0.0], [7.0, 0.0], {"subtype": "segment"}], [[15.0, 9.0], [22.0, 9.0], {"subtype": "segment"}], [[15.0, 0.0], [22.0, 0.0], {"subtype": "segment"}], [[22.0, 0.0], [22.0, 9.0], {"subtype": "segment"}], [[15.0, 9.0], [15.0, 16.0], {"subtype": "segment"}], [[7.0, 0.0], [7.0, 9.0], {"subtype": "segment"}], [[15.0, 16.0], [22.0, 16.0], {"subtype": "segment"}], [[7.0, 25.0], [7.0, 32.0], {"subtype": "segment"}], [[15.0, 25.0], [22.0, 25.0], {"subtype": "segment"}], [[7.0, 9.0], [15.0, 9.0], {"subtype": "segment"}], [[7.0, 9.0], [7.0, 16.0], {"subtype": "segment"}], [[7.0, 16.0], [7.0, 25.0], {"subtype": "segment"}], [[15.0, 0.0], [15.0, 9.0], {"subtype": "segment"}], [[0.0, 9.0], [7.0, 9.0], {"subtype": "segment"}]], "label": [[[0.0, 4.5], "9 cm", "left", {}], [[3.5, 9.0], "7 cm", "above", {}], [[11.0, 32.0], "8 cm", "above", {}], [[7.0, 28.5], "7 cm", "left", {}], [[7.0, 20.5], "9 cm", "left", {}], [[18.5, 25.0], "7 cm", "above", {}], [[7.0, 12.5], "7 cm", "left", {}]]}
D.

{"init": {"range": [[-1.25, 36.0], [-1.0, 23.25]], "scale": [20, 20]}, "line": [[[35.0, 0.0], [35.0, 7.0], {"subtype": "segment"}], [[0.0, 15.0], [0.0, 22.0], {"subtype": "segment"}], [[17.0, 7.0], [26.0, 7.0], {"subtype": "segment"}], [[0.0, 15.0], [8.0, 15.0], {"subtype": "segment"}], [[17.0, 7.0], [17.0, 15.0], {"subtype": "segment"}], [[35.0, 7.0], [35.0, 15.0], {"subtype": "segment"}], [[26.0, 0.0], [26.0, 7.0], {"subtype": "segment"}], [[26.0, 7.0], [26.0, 15.0], {"subtype": "segment"}], [[17.0, 15.0], [26.0, 15.0], {"subtype": "segment"}], [[8.0, 7.0], [8.0, 15.0], {"subtype": "segment"}], [[26.0, 15.0], [35.0, 15.0], {"subtype": "segment"}], [[8.0, 22.0], [17.0, 22.0], {"subtype": "segment"}], [[26.0, 0.0], [35.0, 0.0], {"subtype": "segment"}], [[8.0, 15.0], [8.0, 22.0], {"subtype": "segment"}], [[26.0, 7.0], [35.0, 7.0], {"subtype": "segment"}], [[8.0, 7.0], [17.0, 7.0], {"subtype": "segment"}], [[8.0, 15.0], [17.0, 15.0], {"subtype": "segment"}], [[0.0, 22.0], [8.0, 22.0], {"subtype": "segment"}], [[17.0, 15.0], [17.0, 22.0], {"subtype": "segment"}]], "label": [[[21.5, 15.0], "9 cm", "above", {}], [[8.0, 11.0], "8 cm", "left", {}], [[26.0, 3.5], "7 cm", "left", {}], [[12.5, 22.0], "9 cm", "above", {}], [[4.0, 22.0], "8 cm", "above", {}], [[30.5, 15.0], "9 cm", "above", {}], [[0.0, 18.5], "7 cm", "left", {}]]}

Exercice 2 : Reconnaître un polyèdre - Nommer et compter sommets/faces/arêtes

On considère le solide ci-dessous :

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 500, "height": 350}}, "data": {"Cylinder_1": {"position": {"x": 0.0, "y": -5.0, "z": 20.0}, "rotation": {"x": -1.57079632679, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "GenericRightCylinder", "path": "Triangle_0", "amount": 40}, "Triangle_0": {"type": "Path2D", "path": [{"moveTo": {"x": 0.0, "y": 0.0}}, {"lineTo": {"x": 40.0, "y": 0.0}}, {"lineTo": {"x": 12.0, "y": 28.0}}]}, "Point3D_O": {"type": "Point", "position": {"x": 0.0, "y": -5.0, "z": 20.0}}, "Label_for_Point3D_O": {"type": "Label", "position": {"x": 0.0, "y": -5.0, "z": 20.0}, "text": "O", "position_offset": {"x": 0.0, "y": -4.8, "z": 0.0}}, "Point3D_G": {"type": "Point", "position": {"x": 40.0, "y": -5.0, "z": 20.0}}, "Label_for_Point3D_G": {"type": "Label", "position": {"x": 40.0, "y": -5.0, "z": 20.0}, "text": "G", "position_offset": {"x": 4.762933808225604, "y": -0.5953667260282005, "z": 0.0}}, "Point3D_N": {"type": "Point", "position": {"x": 12.0, "y": -4.9999999998628955, "z": -8.0}}, "Label_for_Point3D_N": {"type": "Label", "position": {"x": 12.0, "y": -4.9999999998628955, "z": -8.0}, "text": "N", "position_offset": {"x": 1.8658470193695444, "y": -0.7774362580493255, "z": -4.353643045195604}}, "Point3D_J": {"type": "Point", "position": {"x": 0.0, "y": 35.0, "z": 20.000000000195865}}, "Label_for_Point3D_J": {"type": "Label", "position": {"x": 0.0, "y": 35.0, "z": 20.000000000195865}, "text": "J", "position_offset": {"x": 0.0, "y": 4.8, "z": 2.6861438787558914e-11}}, "Point3D_H": {"type": "Point", "position": {"x": 40.0, "y": 35.0, "z": 20.000000000195865}}, "Label_for_Point3D_H": {"type": "Label", "position": {"x": 40.0, "y": 35.0, "z": 20.000000000195865}, "text": "H", "position_offset": {"x": 3.612368134593013, "y": 3.160822117768887, "z": 1.7688381215585684e-11}}, "Point3D_V": {"type": "Point", "position": {"x": 12.0, "y": 35.000000000137106, "z": -7.999999999804135}}, "Label_for_Point3D_V": {"type": "Label", "position": {"x": 12.0, "y": 35.000000000137106, "z": -7.999999999804135}, "text": "V", "position_offset": {"x": 1.2413677786677848, "y": 3.6206560211285552, "z": -2.8965248168712363}}}}

On pourra faire pivoter la figure en cliquant dessus.

Quelle est sa nature ?

Combien a-t-il de sommets ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il de faces ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il d'arêtes ?

Les nommer en les séparant par des points-virgule.

Exercice 3 : Connaître les propriétés du prisme droit

En utilisant le prisme droit ci-dessous, répondre aux questions posées.

Est-ce que l'angle \(\widehat{NGE}\) est un angle droit ?

Est-ce que l'angle \(\widehat{ZEG}\) est un angle droit ?

Sélectionner l'arête parallèle à l'arête \([GE]\).

Est-ce que l'angle \(\widehat{NJH}\) est un angle droit ?

Exercice 4 : Nommer et compter sommets/faces/arêtes d'un polyèdre

On considère le solide ci-dessous :

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 500, "height": 350}}, "data": {"Cylinder_1": {"position": {"x": 0.0, "y": -5.0, "z": 20.0}, "rotation": {"x": -1.57079632679, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "GenericRightCylinder", "path": "Triangle_0", "amount": 40}, "Triangle_0": {"type": "Path2D", "path": [{"moveTo": {"x": 0.0, "y": 0.0}}, {"lineTo": {"x": 40.0, "y": 0.0}}, {"lineTo": {"x": 12.0, "y": 28.0}}]}, "Point3D_H": {"type": "Point", "position": {"x": 0.0, "y": -5.0, "z": 20.0}}, "Label_for_Point3D_H": {"type": "Label", "position": {"x": 0.0, "y": -5.0, "z": 20.0}, "text": "H", "position_offset": {"x": 0.0, "y": -4.8, "z": 0.0}}, "Point3D_T": {"type": "Point", "position": {"x": 40.0, "y": -5.0, "z": 20.0}}, "Label_for_Point3D_T": {"type": "Label", "position": {"x": 40.0, "y": -5.0, "z": 20.0}, "text": "T", "position_offset": {"x": 4.762933808225604, "y": -0.5953667260282005, "z": 0.0}}, "Point3D_W": {"type": "Point", "position": {"x": 12.0, "y": -4.9999999998628955, "z": -8.0}}, "Label_for_Point3D_W": {"type": "Label", "position": {"x": 12.0, "y": -4.9999999998628955, "z": -8.0}, "text": "W", "position_offset": {"x": 1.8658470193695444, "y": -0.7774362580493255, "z": -4.353643045195604}}, "Point3D_I": {"type": "Point", "position": {"x": 0.0, "y": 35.0, "z": 20.000000000195865}}, "Label_for_Point3D_I": {"type": "Label", "position": {"x": 0.0, "y": 35.0, "z": 20.000000000195865}, "text": "I", "position_offset": {"x": 0.0, "y": 4.8, "z": 2.6861438787558914e-11}}, "Point3D_X": {"type": "Point", "position": {"x": 40.0, "y": 35.0, "z": 20.000000000195865}}, "Label_for_Point3D_X": {"type": "Label", "position": {"x": 40.0, "y": 35.0, "z": 20.000000000195865}, "text": "X", "position_offset": {"x": 3.612368134593013, "y": 3.160822117768887, "z": 1.7688381215585684e-11}}, "Point3D_A": {"type": "Point", "position": {"x": 12.0, "y": 35.000000000137106, "z": -7.999999999804135}}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": 12.0, "y": 35.000000000137106, "z": -7.999999999804135}, "text": "A", "position_offset": {"x": 1.2413677786677848, "y": 3.6206560211285552, "z": -2.8965248168712363}}}}

On pourra faire pivoter la figure en cliquant dessus.

Combien a-t-il de sommets ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il de faces ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il d'arêtes ?

Les nommer en les séparant par des points-virgule.

Exercice 5 : Reconnaître le patron d'un polyèdre (cube, pavé droit, prisme droit)

Voici le patron d'un solide tel que \( ZL = 2 cm \) ; \( LB = 3 cm \) ; \( ZB = 4 cm \) et \( FE = 6 cm \).

{"init": {"range": [[-4.5, 2.0], [-1.5, 5.5]], "scale": [80, 80]}, "line": [[[-2.69, 0.06], [-2.81, -0.06]], [[-2.75, 0.06], [-2.87, -0.06]], [[-2.63, 0.06], [-2.75, -0.06]], [[-3.5, 0.0], [-2.0, 0.0], {"subtype": "segment"}], [[-1.3319999999999999, -0.222], [-1.468, -0.278]], [[-1.38, -0.202], [-1.516, -0.258]], [[-1.2839999999999998, -0.24200000000000002], [-1.42, -0.29800000000000004]], [[-2.0, 0.0], [-0.8, -0.5], {"subtype": "segment"}], [[-0.388, -0.198], [-0.41200000000000003, -0.302]], [[-0.8, -0.5], [0.0, 0.0], {"subtype": "segment"}], [[0.54, 0.04], [0.46, -0.04]], [[0.0, 0.0], [1.0, 0.0], {"subtype": "segment"}], [[1.16, 1.84], [0.84, 2.16]], [[1.16, 2.0], [0.84, 2.3200000000000003]], [[1.0, 0.0], [1.0, 4.0], {"subtype": "segment"}], [[0.54, 4.04], [0.46, 3.96]], [[1.0, 4.0], [0.0, 4.0], {"subtype": "segment"}], [[-0.452, 4.238], [-0.34800000000000003, 4.262]], [[0.0, 4.0], [-0.8, 4.5], {"subtype": "segment"}], [[-1.428, 4.182], [-1.3719999999999999, 4.318]], [[-1.38, 4.202], [-1.3239999999999998, 4.337999999999999]], [[-1.476, 4.162000000000001], [-1.42, 4.298]], [[-0.8, 4.5], [-2.0, 4.0], {"subtype": "segment"}], [[-2.69, 4.06], [-2.81, 3.94]], [[-2.75, 4.06], [-2.87, 3.94]], [[-2.63, 4.06], [-2.75, 3.94]], [[-2.0, 4.0], [-3.5, 4.0], {"subtype": "segment"}], [[-3.34, 1.84], [-3.66, 2.16]], [[-3.34, 2.0], [-3.66, 2.3200000000000003]], [[-3.5, 4.0], [-3.5, 0.0], {"subtype": "segment"}], [[-0.92, 0.08], [-1.08, -0.08]], [[-1.08, 0.08], [-0.92, -0.08]], [[-2.0, 0.0], [0.0, 0.0], {"subtype": "segment"}], [[-0.92, 4.08], [-1.08, 3.92]], [[-1.08, 4.08], [-0.92, 3.92]], [[-2.0, 4.0], [0.0, 4.0], {"subtype": "segment"}], [[-1.84, 1.84], [-2.16, 2.16]], [[-1.84, 2.0], [-2.16, 2.3200000000000003]], [[-2.0, 4.0], [-2.0, 0.0], {"subtype": "segment"}], [[0.16, 1.84], [-0.16, 2.16]], [[0.16, 2.0], [-0.16, 2.3200000000000003]], [[0.0, 4.0], [0.0, 0.0], {"subtype": "segment"}]], "label": [[[0.0, 0.0], "O", "below right", {}], [[1.0, 0.0], "F", "below right", {}], [[1.0, 4.0], "E", "above right", {}], [[0.0, 4.0], "Z", "above right", {}], [[-0.8, 4.5], "L", "above", {}], [[-2.0, 4.0], "B", "above left", {}], [[-3.5, 4.0], "I", "above left", {}], [[-3.5, 0.0], "M", "below left", {}], [[-2.0, 0.0], "G", "below left", {}], [[-0.8, -0.5], "V", "below", {}]]}

1. Construire ce solide
2. De quel type de solide s'agit-il ?

Revenir au choix du niveau