Similitudes et transformations géométriques — Quiz interactif pour Terminale Math

Question 1 sur 10 10:00

[{"id":9283,"question":"Quelle transformation géométrique est une similitude directe ?","option_a":"Une translation","option_b":"Une rotation suivie d'une homothétie","option_c":"Une réflexion","option_d":"Une projection orthogonale","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une similitude directe combine une rotation et une homothétie (éventuellement une translation), ce qui conserve les angles et les rapports de distances.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9284,"question":"Dans le plan complexe, une similitude est représentée par l'application z ↦ (2 + 3i)z + 1. Quel est le rapport de cette similitude ?","option_a":"2","option_b":"3","option_c":"√13","option_d":"5","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le rapport d'une similitude est donné par le module du coefficient multiplicatif (ici |2 + 3i| = √(2² + 3²) = √13).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9285,"question":"Une similitude indirecte conserve l'orientation du plan.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une similitude indirecte renverse l'orientation (exemple : une réflexion), contrairement à une similitude directe qui la conserve.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9286,"question":"Quelle est l'image d'un point d'affixe z = 1 + i par la similitude z ↦ (1 - i)z + 2i ?","option_a":"3 + i","option_b":"1 + 3i","option_c":"2 + 2i","option_d":"0 + 2i","option_e":"","option_f":"","bonne_reponse":"A","explication":"Calcul : (1 - i)(1 + i) + 2i = (1 - i²) + 2i = (1 + 1) + 2i = 2 + 2i. L'image est donc 2 + 2i.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9287,"question":"Toute similitude est une bijection du plan.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une similitude est une transformation bijective car elle admet une application réciproque (inverse de la similitude).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9288,"question":"Quel est l'angle de rotation d'une similitude directe de rapport 2 et d'argument π\/4 ?","option_a":"π\/4","option_b":"π\/2","option_c":"π","option_d":"2π","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'argument du coefficient multiplicatif (ici π\/4) donne directement l'angle de rotation de la similitude.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9289,"question":"Une homothétie est-elle une similitude ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une homothétie est un cas particulier de similitude où l'angle de rotation est nul (argument du coefficient multiplicatif = 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9290,"question":"Quelle est la nature de la transformation z ↦ -z ?","option_a":"Rotation de π","option_b":"Homothétie de rapport -1","option_c":"Symétrie centrale","option_d":"Translation","option_e":"","option_f":"","bonne_reponse":"A","explication":"La transformation z ↦ -z est une rotation de π radians (180°) autour de l'origine, donc une similitude directe de rapport 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9291,"question":"Le produit de deux similitudes est-il toujours une similitude ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le produit de deux similitudes (composées) est une similitude, car la composition de transformations bijectives est bijective et conserve les rapports de distances.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":9292,"question":"Quelle similitude transforme le point A(1, 0) en B(0, 1) et conserve l'orientation ?","option_a":"Rotation de π\/2 autour de l'origine","option_b":"Homothétie de centre O et rapport 1","option_c":"Translation de vecteur (0,1)","option_d":"Réflexion d'axe y = x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La rotation de π\/2 autour de l'origine transforme (1,0) en (0,1) et conserve l'orientation (similitude directe).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]