Géométrie dans l'espace

Pour aller plus loin (Ancien programme) - Mathématiques 5e

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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"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": 36.0, "y": 54.0, "z": 24.0}, "point_render": {"color": 204}, "label_render": {"color": 204}}, "Label_for_Point3D_97": {"type": "Label", "position": {"x": 36.0, "y": 54.0, "z": 24.0}, "text": "M", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}, "label_render": {"color": 204}}, "Point3D_98": {"type": "Point", "position": {"x": 36.0, "y": 0.0, "z": 24.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Point3D_99": {"type": "Point", "position": {"x": 36.0, "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": 24.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Segment3D_101": {"type": "Segment", "pt1": {"x": 36.0, "y": 54.0, "z": 24.0}, "pt2": {"x": 36.0, "y": 0.0, "z": 24.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}, "Segment3D_102": {"type": "Segment", "pt1": {"x": 36.0, "y": 54.0, "z": 24.0}, "pt2": {"x": 36.0, "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": 36.0, "y": 54.0, "z": 24.0}, "pt2": {"x": 0.0, "y": 54.0, "z": 24.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}}}

Exercice 2 : Lire les coordonnées géographique d'une ville sur terre à partir des parallèles et méridiens

Sur le schéma ci-dessous, le point D symbolise la ville d'Elx (Espagne).

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"Label_for_Point3D_D": {"type": "Label", "position": {"x": -30.64177772475912, "y": 25.71150438746157, "z": 0.0}, "text": "D", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 16711680}}}}

On pourra faire pivoter la figure en cliquant dessus.

Légende

Méridien d'origine Greenwich 0°
Méridiens (espacés de 20°)
Équateur 0°
Parallèles (espacés de 20°)
N : Pôle nord
S : Pôle sud
D : Ville d'Elx

Donner les coordonnées géographique de la ville d'Elx (Espagne).
Par exemple:

Pour la ville de Tema (Ghana) on donnera une réponse de la forme \(\left(6° N; 0°\right)\)

Pour la ville de Punto Fijo (Vénézuéla) on donnera une réponse de la forme \(\left(12° N; 70° O\right)\)

Exercice 3 : Calcul du volume d'une boule

Soit une boule de \(12 cm\) de rayon.

Calculer son volume.
On donnera le résultat exact.

Exercice 4 : Lire les coordonnées géographiques d'une ville sur terre à partir des angles

Sur le schéma ci-dessous, le point D symbolise la ville de Managua (Nicaragua).

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"Point3D_E": {"type": "Point", "position": {"x": -2.7902589497650094, "y": 0.0, "z": -39.90256201039297}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": -2.7902589497650094, "y": 0.0, "z": -39.90256201039297}, "text": "E", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_A": {"type": "Point", "position": {"x": -39.12590402935223, "y": 8.316467632710374, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": -39.12590402935223, "y": 8.316467632710374, "z": 0.0}, "text": "A", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_D": {"type": "Point", "position": {"x": -2.729285097138672, "y": 8.316467632710374, "z": -39.030595293597784}, "point_render": {"color": 16711680, "size": 128}, "label_render": {"color": 16711680}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": -2.729285097138672, "y": 8.316467632710374, "z": -39.030595293597784}, "text": "D", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 16711680}}}}

On pourra faire pivoter la figure en cliquant dessus.

Légende

Méridien d'origine Greenwich 0°
Méridien du point D
Équateur 0°
Parallèle du point D
N : Pôle nord
S : Pôle sud
C : Centre de la terre
D : Ville de Managua

Sachant que \( \widehat{ABD} = 86 ° \), \( \widehat{ECD} = 12 ° \), \( \widehat{SCE} = 90 ° \) et \( \widehat{DCN} = 78 ° \), donner les coordonnées géographique de la ville de Managua (Nicaragua).
Par exemple:

Pour la ville d'Oum el Bouaghi (Algérie) on donnera une réponse de la forme \(\left(36° N; 7° E\right)\)

Pour la ville de Recife (Brésil) on donnera une réponse de la forme \(\left(8° S; 35° O\right)\)

Exercice 5 : Calcul d'un rayon à partir de l'aire d'une sphère

Soit une sphère d'aire \(100\:cm^{2}\). Calculer son rayon.

On donnera le résultat sous la forme d'une valeur exacte avec \( \sqrt{\pi } \).

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