Parallélogrammes & Obtention

Le plan - Mathématiques 5e

Exercice 1 : Reconnaître un rectangle par ses côtés et ses angles

Parmi les quadrilatères suivants, lesquels sont des rectangles ?

1.

{"path": [[[[-5.49726061320878, 4.333604244790176], [0.0, 0.0], [2.0003016851039663, 4.582370365713604], [-3.4969589281048137, 8.91597461050378], [-5.49726061320878, 4.333604244790176]]]], "init": {"range": [[-6.49726061320878, 3.0003016851039663], [-1.0, 9.91597461050378]], "scale": [30, 30]}, "line": [[[-2.514562095699653, 2.1945082264050546], [-2.9826985175091267, 2.1390960183851213]], [[-2.645449253157005, 2.2976892798524395], [-3.1135856749664788, 2.2422770718325062]], [[-2.383674938242301, 2.0913271729576697], [-2.8518113600517747, 2.0359149649377364]], [[-5.49726061320878, 4.333604244790176], [0.0, 0.0], {"subtype": "segment"}], [[0.914080724577226, 2.51061054681313], [1.0862209605267403, 2.0717598189004742]], [[0.7807254785956553, 2.205115064882045], [1.219576206508311, 2.3772553008315596]], [[0.0, 0.0], [2.0003016851039663, 4.582370365713604], {"subtype": "segment"}], [[-0.9823968324051606, 6.721466384098725], [-0.5142604105956867, 6.776878592118659]], [[-0.8515096749478088, 6.618285330651339], [-0.38337325313833476, 6.673697538671274]], [[-1.1132839898625126, 6.82464743754611], [-0.6451475680530385, 6.880059645566044]], [[2.0003016851039663, 4.582370365713604], [-3.4969589281048137, 8.91597461050378], {"subtype": "segment"}], [[-4.41103965268204, 6.405364063690651], [-4.5831798886315545, 6.844214791603306]], [[-4.27768440670047, 6.7108595456217355], [-4.716535134613125, 6.538719309672221]], [[-3.4969589281048137, 8.91597461050378], [-5.49726061320878, 4.333604244790176], {"subtype": "segment"}]], "label": [[[-5.49726061320878, 4.333604244790176], "D", "left", {}], [[0.0, 0.0], "E", "below", {}], [[2.0003016851039663, 4.582370365713604], "F", "right", {}], [[-3.4969589281048137, 8.91597461050378], "G", "above", {}]]}
2.

{"path": [[[[-5.157071942979224, 4.733350713282979], [0.0, 0.0], [6.08573663136383, 6.6305210695447165], [0.9286646883846064, 11.363871782827696], [-5.157071942979224, 4.733350713282979]]], [[[0.0, 0.0], [-0.3683622816413731, 0.33809647952021277], [-0.030265802121160312, 0.7064587611615859], [0.33809647952021277, 0.36836228164137313], [0.0, 0.0]]], [[[-5.157071942979224, 4.733350713282979], [-4.788709661337851, 4.395254233762767], [-4.450613181817638, 4.76361651540414], [-4.818975463459012, 5.101712994924353], [-5.157071942979224, 4.733350713282979]]], [[[0.9286646883846064, 11.363871782827696], [0.5905682088643937, 10.995509501186323], [0.9589304905057667, 10.65741302166611], [1.2970269700259796, 11.025775303307483], [0.9286646883846064, 11.363871782827696]]], [[[6.08573663136383, 6.6305210695447165], [5.717374349722458, 6.96861754906493], [5.379277870202245, 6.600255267423557], [5.747640151843618, 6.262158787903344], [6.08573663136383, 6.6305210695447165]]]], "init": {"range": [[-6.157071942979224, 7.08573663136383], [-1.0, 12.363871782827696]], "scale": [30, 30]}, "line": [[[-5.157071942979224, 4.733350713282979], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [6.08573663136383, 6.6305210695447165], {"subtype": "segment"}], [[6.08573663136383, 6.6305210695447165], [0.9286646883846064, 11.363871782827696], {"subtype": "segment"}], [[0.9286646883846064, 11.363871782827696], [-5.157071942979224, 4.733350713282979], {"subtype": "segment"}]], "label": [[[-2.984251746913867, 1.9246406186718419], "7"], [[3.484903053651563, 2.909544759348103], "9"], [[3.9129164352984738, 9.439231164155855], "7"], [[-2.5562383652669567, 8.454327023479593], "9"], [[-5.157071942979224, 4.733350713282979], "C", "left", {}], [[0.0, 0.0], "D", "below", {}], [[6.08573663136383, 6.6305210695447165], "E", "right", {}], [[0.9286646883846064, 11.363871782827696], "F", "above", {}]]}
3.

{"path": [[[[-1.1761064417418838, 8.922823187629197], [0.0, 0.0], [4.900404172264108, 0.9918830316405923], [2.797461714126764, 13.41863558481245], [-1.1761064417418838, 8.922823187629197]]]], "init": {"range": [[-2.176106441741884, 5.900404172264108], [-1.0, 14.41863558481245]], "scale": [30, 30]}, "line": [[[-0.40103600551221824, 4.317953876298167], [-0.7750704362296656, 4.60486931133103]], [[-0.4445955033545102, 4.648428809173322], [-0.7315109383873737, 4.274394378455875]], [[-1.1761064417418838, 8.922823187629197], [0.0, 0.0], {"subtype": "segment"}], [[2.5804918752710604, 0.6923597593764099], [2.3199122969930475, 0.29952327226418246]], [[2.4171378589235, 0.6592955321678562], [2.1565582806454873, 0.2664590450556288]], [[0.0, 0.0], [4.900404172264108, 0.9918830316405923], {"subtype": "segment"}], [[3.6567935742759468, 7.341780436506271], [4.041072312114926, 7.068738179946772]], [[3.6846026945958164, 7.1774501879066515], [4.068881432434795, 6.904407931347152]], [[3.628984453956077, 7.506110685105891], [4.013263191795056, 7.2330684285463915]], [[4.900404172264108, 0.9918830316405923], [2.797461714126764, 13.41863558481245], {"subtype": "segment"}], [[0.8251841037508431, 10.935473954924257], [0.7961711686340375, 11.405984817517387]], [[2.797461714126764, 13.41863558481245], [-1.1761064417418838, 8.922823187629197], {"subtype": "segment"}]], "label": [[[-1.1761064417418838, 8.922823187629197], "D", "above left", {}], [[0.0, 0.0], "E", "below", {}], [[4.900404172264108, 0.9918830316405923], "F", "below right", {}], [[2.797461714126764, 13.41863558481245], "G", "above", {}]]}
4.

{"path": [[[[-4.65671487111079, 5.22637604934582], [0.0, 0.0], [6.719626349158911, 5.987204834285301], [2.0629114780481226, 11.21358088363112], [-4.65671487111079, 5.22637604934582]]], [[[0.0, 0.0], [-0.3326224907936278, 0.37331257495327286], [0.040690084159645035, 0.7059350657469006], [0.37331257495327286, 0.33262249079362777], [0.0, 0.0]]], [[[-4.65671487111079, 5.22637604934582], [-4.324092380317161, 4.853063474392547], [-3.9507798053638887, 5.185685965186175], [-4.283402296157516, 5.558998540139448], [-4.65671487111079, 5.22637604934582]]], [[[2.0629114780481226, 11.21358088363112], [1.6895989030948497, 10.880958392837494], [2.0222213938884774, 10.507645817884221], [2.3955339688417503, 10.840268308677848], [2.0629114780481226, 11.21358088363112]]], [[[6.719626349158911, 5.987204834285301], [6.387003858365284, 6.3605174092385734], [6.013691283412011, 6.027894918444946], [6.346313774205639, 5.654582343491673], [6.719626349158911, 5.987204834285301]]]], "init": {"range": [[-5.65671487111079, 7.719626349158911], [-1.0, 12.21358088363112]], "scale": [30, 30]}, "line": [[[-1.9989210715401746, 2.5941993187317425], [-2.6577937995706153, 2.6321767306140775]], [[-4.65671487111079, 5.22637604934582], [0.0, 0.0], {"subtype": "segment"}], [[3.3788018805206232, 3.3230387811578708], [3.340824468638288, 2.66416605312743]], [[3.0303768105642352, 3.012591123083818], [3.689249538594676, 2.9746137112014828]], [[0.0, 0.0], [6.719626349158911, 5.987204834285301], {"subtype": "segment"}], [[4.061832549588297, 8.619381564899378], [4.7207052776187375, 8.581404153017044]], [[6.719626349158911, 5.987204834285301], [2.0629114780481226, 11.21358088363112], {"subtype": "segment"}], [[-1.315890402472501, 7.89054210247325], [-1.2779129905901656, 8.549414830503691]], [[-0.967465332516113, 8.200989760547303], [-1.6263380605465536, 8.238967172429637]], [[2.0629114780481226, 11.21358088363112], [-4.65671487111079, 5.22637604934582], {"subtype": "segment"}]], "label": [[[-4.65671487111079, 5.22637604934582], "H", "left", {}], [[0.0, 0.0], "I", "below", {}], [[6.719626349158911, 5.987204834285301], "J", "right", {}], [[2.0629114780481226, 11.21358088363112], "K", "above", {}]]}

Exercice 2 : Reconnaître un carré par ses côtés et ses angles

Parmi les quadrilatères suivants, lesquels sont des carrés ?

1.

{"path": [[[[-8.600073366674463, 2.6530620211025138], [0.0, 0.0], [2.6530620211025138, 8.600073366674463], [-5.9470113455719495, 11.253135387776977], [-8.600073366674463, 2.6530620211025138]]], [[[0.0, 0.0], [-0.47778185370413684, 0.1473923345056952], [-0.3303895191984416, 0.6251741882098321], [0.1473923345056952, 0.47778185370413684], [0.0, 0.0]]], [[[-8.600073366674463, 2.6530620211025138], [-8.122291512970326, 2.5056696865968187], [-7.974899178464631, 2.9834515403009556], [-8.452681032168769, 3.1308438748066507], [-8.600073366674463, 2.6530620211025138]]], [[[-5.9470113455719495, 11.253135387776977], [-6.094403680077645, 10.77535353407284], [-5.616621826373508, 10.627961199567146], [-5.469229491867813, 11.105743053271283], [-5.9470113455719495, 11.253135387776977]]], [[[2.6530620211025138, 8.600073366674463], [2.1752801673983773, 8.74746570118016], [2.0278878328926817, 8.269683847476022], [2.5056696865968187, 8.122291512970326], [2.6530620211025138, 8.600073366674463]]]], "init": {"range": [[-9.600073366674463, 3.6530620211025138], [-1.0, 12.253135387776977]], "scale": [30, 30]}, "line": [[[-8.600073366674463, 2.6530620211025138], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [2.6530620211025138, 8.600073366674463], {"subtype": "segment"}], [[2.6530620211025138, 8.600073366674463], [-5.9470113455719495, 11.253135387776977], {"subtype": "segment"}], [[-5.9470113455719495, 11.253135387776977], [-8.600073366674463, 2.6530620211025138], {"subtype": "segment"}]], "label": [[[-4.476907484744066, 0.7531927861062927], "9"], [[1.899869234996221, 4.123165881930397], "9"], [[-1.4701038608278836, 10.499942601670684], "9"], [[-7.846880580568171, 7.12996950584658], "9"], [[-8.600073366674463, 2.6530620211025138], "A", "below left", {}], [[0.0, 0.0], "B", "below right", {}], [[2.6530620211025138, 8.600073366674463], "C", "above right", {}], [[-5.9470113455719495, 11.253135387776977], "D", "above left", {}]]}
2.

{"path": [[[[3.5330379497610047, -7.177579177239952], [0.0, 0.0], [-7.177579177239952, -3.5330379497610047], [-3.644541227478948, -10.710617127000958], [3.5330379497610047, -7.177579177239952]]], [[[0.0, 0.0], [0.2208148718600628, -0.448598698577497], [-0.22778382671743422, -0.6694135704375598], [-0.448598698577497, -0.2208148718600628], [0.0, 0.0]]], [[[3.5330379497610047, -7.177579177239952], [3.312223077900942, -6.728980478662455], [2.863624379323445, -6.9497953505225185], [3.084439251183508, -7.398394049100015], [3.5330379497610047, -7.177579177239952]]], [[[-3.644541227478948, -10.710617127000958], [-3.1959425289014507, -10.489802255140894], [-3.4167574007615134, -10.041203556563397], [-3.8653560993390106, -10.26201842842346], [-3.644541227478948, -10.710617127000958]]], [[[-7.177579177239952, -3.5330379497610047], [-6.95676430537989, -3.981636648338502], [-6.508165606802393, -3.7608217764784393], [-6.728980478662455, -3.312223077900942], [-7.177579177239952, -3.5330379497610047]]]], "init": {"range": [[-8.177579177239952, 4.533037949761005], [-11.710617127000958, 1.0]], "scale": [30, 30]}, "line": [[[1.4094984039804705, -3.4673048810373444], [2.123539545780534, -3.710274296202608]], [[3.5330379497610047, -7.177579177239952], [0.0, 0.0], {"subtype": "segment"}], [[-3.710274296202608, -2.123539545780534], [-3.4673048810373444, -1.4094984039804705]], [[0.0, 0.0], [-7.177579177239952, -3.5330379497610047], {"subtype": "segment"}], [[-5.0540396314594185, -7.243312245963613], [-5.768080773259482, -7.00034283079835]], [[-7.177579177239952, -3.5330379497610047], [-3.644541227478948, -10.710617127000958], {"subtype": "segment"}], [[0.06573306872366017, -8.587077581220424], [-0.17723634644160297, -9.301118723020487]], [[-3.644541227478948, -10.710617127000958], [3.5330379497610047, -7.177579177239952], {"subtype": "segment"}]], "label": [[[3.5330379497610047, -7.177579177239952], "P", "right", {}], [[0.0, 0.0], "Q", "above", {}], [[-7.177579177239952, -3.5330379497610047], "S", "left", {}], [[-3.644541227478948, -10.710617127000958], "T", "below", {}]]}
3.

{"path": [[[[-8.57863972191678, -2.721569495998883], [0.0, 0.0], [-2.721569495998883, 8.57863972191678], [-11.300209217915661, 5.857070225917896], [-8.57863972191678, -2.721569495998883]]], [[[0.0, 0.0], [-0.4765910956620433, -0.15119830533327128], [-0.6277894009953146, 0.325392790328772], [-0.15119830533327128, 0.4765910956620433], [0.0, 0.0]]], [[[-8.57863972191678, -2.721569495998883], [-8.102048626254735, -2.5703711906656115], [-8.253246931588007, -2.0937800950035683], [-8.72983802725005, -2.24497840033684], [-8.57863972191678, -2.721569495998883]]], [[[-11.300209217915661, 5.857070225917896], [-11.14901091258239, 5.380479130255853], [-10.672419816920346, 5.531677435589124], [-10.823618122253619, 6.008268531251167], [-11.300209217915661, 5.857070225917896]]], [[[-2.721569495998883, 8.57863972191678], [-3.198160591660926, 8.427441416583507], [-3.0469622863276546, 7.9508503209214645], [-2.5703711906656115, 8.102048626254735], [-2.721569495998883, 8.57863972191678]]]], "init": {"range": [[-12.300209217915661, 1.0], [-3.721569495998883, 9.57863972191678]], "scale": [30, 30]}, "line": [[[-8.57863972191678, -2.721569495998883], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-2.721569495998883, 8.57863972191678], {"subtype": "segment"}], [[-2.721569495998883, 8.57863972191678], [-11.300209217915661, 5.857070225917896], {"subtype": "segment"}], [[-11.300209217915661, 5.857070225917896], [-8.57863972191678, -2.721569495998883], {"subtype": "segment"}]], "label": [[[-4.1078818945584645, -1.9326940627938933], "9"], [[-0.7888754332049897, 4.470757827358315], "9"], [[-7.192327323357198, 7.789764288711789], "9"], [[-10.511333784710672, 1.3863123985595809], "9"], [[-8.57863972191678, -2.721569495998883], "K", "below left", {}], [[0.0, 0.0], "L", "below right", {}], [[-2.721569495998883, 8.57863972191678], "M", "above right", {}], [[-11.300209217915661, 5.857070225917896], "N", "above left", {}]]}
4.

{"path": [[[[-0.9259857373486244, 5.928115249742276], [0.0, 0.0], [8.892172874613413, 1.3889786060229365], [6.242731553307136, 9.480024934215283], [-0.9259857373486244, 5.928115249742276]]], [[[0.0, 0.0], [-0.07716547811238536, 0.49400960414518963], [0.4168441260328042, 0.571175082257575], [0.4940096041451896, 0.07716547811238536], [0.0, 0.0]]]], "init": {"range": [[-1.9259857373486244, 9.892172874613413], [-1.0, 10.480024934215283]], "scale": [30, 30]}, "line": [[[-0.9259857373486244, 5.928115249742276], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [8.892172874613413, 1.3889786060229365], {"subtype": "segment"}], [[8.892172874613413, 1.3889786060229365], [6.242731553307136, 9.480024934215283], {"subtype": "segment"}], [[6.242731553307136, 9.480024934215283], [-0.9259857373486244, 5.928115249742276], {"subtype": "segment"}]], "label": [[[-1.0558043936485397, 2.8714590511362754], "6"], [[4.538685011041569, 0.10167777803724076], "9"], [[8.13766001527747, 5.621218302105461], "9"], [[2.391993354055941, 8.241696294141226], "8"], [[-0.9259857373486244, 5.928115249742276], "D", "left", {}], [[0.0, 0.0], "E", "below left", {}], [[8.892172874613413, 1.3889786060229365], "F", "below right", {}], [[6.242731553307136, 9.480024934215283], "G", "above right", {}]]}

Exercice 3 : Compléter le 4e point de la figure pour former un parallélogramme

Ajouter le point D au dessin pour que ABCD soit un parallélogramme, puis compléter le parallélogramme.

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Exercice 4 : Démontrer qu'un quadrilatère est un parallélogramme

Voici le quadrilatère \(PQRS\) :

On sait que \[(PQ) // (RS)\] \[PQ = RS = 3,8 m\]
Compléter la démonstration pour montrer que le quadrilatère \(PQRS\) est un parallélogramme.

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Exercice 5 : Construction de parallélogrammes à partir d'une symétrie centrale

Tracer sur papier un triangle \(HTO\), et placer un point \(S\) à l'extérieur de ce triangle.
Construire les points \(E\), \(M\) et \(G\), respectivement symétriques des points \(H\), \(T\) et \(O\) par la symétrie centrale de centre \(S\).
Citer tous les parallélogrammes présents dans la figure.
On écrira la réponse sous la forme d'une liste séparée par des points-virgule (;).

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