Le plan - 5e
Parallélogrammes & Obtention
Exercice 1 : Construction de parallélogrammes à partir d'une symétrie centrale
Tracer sur papier un triangle \(MFD\), et placer un point
\(G\) à l'extérieur de ce triangle.
Construire les points \(T\), \(P\) et \(I\), respectivement symétriques des points \(M\), \(F\) et \(D\) par la symétrie centrale de centre \(G\).
Citer tous les parallélogrammes présents dans la figure.
On écrira la réponse sous la forme d'une liste séparée par des points-virgule (;).
Construire les points \(T\), \(P\) et \(I\), respectivement symétriques des points \(M\), \(F\) et \(D\) par la symétrie centrale de centre \(G\).
Citer tous les parallélogrammes présents dans la figure.
On écrira la réponse sous la forme d'une liste séparée par des points-virgule (;).
Exercice 2 : Reconnaître un carré par ses côtés et ses angles
Parmi les quadrilatères suivants, lesquels sont des carrés ?
- 1.
{"path": [[[[-7.6836452377807944, -2.2274639974483357], [0.0, 0.0], [-4.652986090191089, 6.507801890384206], [-12.336631327971883, 4.28033789293587], [-7.6836452377807944, -2.2274639974483357]]]], "init": {"range": [[-13.336631327971883, 1.0], [-3.2274639974483357, 7.507801890384206]], "scale": [30, 30]}, "line": [[[-3.659949910879315, -0.7833616908831968], [-4.023695326901479, -1.4441023065651388]], [[-7.6836452377807944, -2.2274639974483357], [0.0, 0.0], {"subtype": "segment"}], [[-2.6985141813273894, 3.315727286010474], [-1.9544719088636995, 3.1920746043737322]], [[0.0, 0.0], [-4.652986090191089, 6.507801890384206], {"subtype": "segment"}], [[-8.676681417092567, 5.0636995838190675], [-8.312936001070405, 5.724440199501009]], [[-4.652986090191089, 6.507801890384206], [-12.336631327971883, 4.28033789293587], {"subtype": "segment"}], [[-9.638117146644493, 0.964610606925397], [-10.382159419108184, 1.0882632885621382]], [[-12.336631327971883, 4.28033789293587], [-7.6836452377807944, -2.2274639974483357], {"subtype": "segment"}]], "label": [[[-7.6836452377807944, -2.2274639974483357], "M", "below", {}], [[0.0, 0.0], "N", "right", {}], [[-4.652986090191089, 6.507801890384206], "O", "above", {}], [[-12.336631327971883, 4.28033789293587], "P", "left", {}]]}
- 2.
{"path": [[[[0.8158728850962083, 4.932986056676502], [0.0, 0.0], [4.932986056676502, -0.8158728850962083], [5.748858941772711, 4.117113171580294], [0.8158728850962083, 4.932986056676502]]], [[[0.0, 0.0], [0.08158728850962083, 0.49329860566765027], [0.5748858941772711, 0.41171131715802944], [0.49329860566765027, -0.08158728850962083], [0.0, 0.0]]], [[[0.8158728850962083, 4.932986056676502], [0.7342855965865874, 4.439687451008852], [1.2275842022542378, 4.358100162499231], [1.3091714907638585, 4.851398768166882], [0.8158728850962083, 4.932986056676502]]], [[[5.748858941772711, 4.117113171580294], [5.25556033610506, 4.198700460089915], [5.17397304759544, 3.7054018544222647], [5.66727165326309, 3.6238145659126437], [5.748858941772711, 4.117113171580294]]], [[[4.932986056676502, -0.8158728850962083], [5.014573345186124, -0.32257427942855804], [4.521274739518473, -0.24098699091893722], [4.439687451008852, -0.7342855965865874], [4.932986056676502, -0.8158728850962083]]]], "init": {"range": [[-1.0, 6.748858941772711], [-1.8158728850962083, 5.932986056676502]], "scale": [30, 30]}, "line": [[[0.8158728850962083, 4.932986056676502], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [4.932986056676502, -0.8158728850962083], {"subtype": "segment"}], [[4.932986056676502, -0.8158728850962083], [5.748858941772711, 4.117113171580294], {"subtype": "segment"}], [[5.748858941772711, 4.117113171580294], [0.8158728850962083, 4.932986056676502], {"subtype": "segment"}]], "label": [[[-0.18402188425307608, 2.5643977745497963], "5"], [[2.368588282126706, -0.9998947693492843], "5"], [[5.932880826025786, 1.5527153970304979], "5"], [[3.380270659646005, 5.117007940929578], "5"], [[0.8158728850962083, 4.932986056676502], "M", "above left", {}], [[0.0, 0.0], "N", "below left", {}], [[4.932986056676502, -0.8158728850962083], "O", "below right", {}], [[5.748858941772711, 4.117113171580294], "P", "above right", {}]]}
- 3.
{"path": [[[[3.3048338851510244, 6.1707433094852995], [0.0, 0.0], [5.837125188927153, 1.3886934610604846], [9.141959074078178, 7.559436770545784], [3.3048338851510244, 6.1707433094852995]]]], "init": {"range": [[-1.0, 10.141959074078178], [-1.0, 8.559436770545785]], "scale": [30, 30]}, "line": [[[3.3048338851510244, 6.1707433094852995], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [5.837125188927153, 1.3886934610604846], {"subtype": "segment"}], [[5.837125188927153, 1.3886934610604846], [9.141959074078178, 7.559436770545784], {"subtype": "segment"}], [[9.141959074078178, 7.559436770545784], [3.3048338851510244, 6.1707433094852995], {"subtype": "segment"}]], "label": [[[1.1234960874767723, 3.3686431306127376], "7"], [[3.0574309762092113, 0.11063826515446185], "6"], [[8.018462986601406, 4.190793639933046], "7"], [[6.084528097868966, 7.448798505391323], "6"], [[3.3048338851510244, 6.1707433094852995], "S", "above left", {}], [[0.0, 0.0], "T", "below left", {}], [[5.837125188927153, 1.3886934610604846], "V", "below right", {}], [[9.141959074078178, 7.559436770545784], "W", "above right", {}]]}
- 4.
{"path": [[[[4.427087042913458, 2.323983716480771], [0.0, 0.0], [2.323983716480771, -4.427087042913458], [6.751070759394229, -2.103103326432687], [4.427087042913458, 2.323983716480771]]], [[[0.0, 0.0], [0.4427087042913458, 0.23239837164807708], [0.6751070759394229, -0.2103103326432687], [0.23239837164807708, -0.4427087042913458], [0.0, 0.0]]], [[[4.427087042913458, 2.323983716480771], [3.984378338622112, 2.0915853448326938], [4.216776710270189, 1.6488766405413477], [4.6594854145615345, 1.8812750121894248], [4.427087042913458, 2.323983716480771]]], [[[6.751070759394229, -2.103103326432687], [6.518672387746151, -1.6603946221413415], [6.075963683454805, -1.8927929937894186], [6.308362055102883, -2.335501698080764], [6.751070759394229, -2.103103326432687]]], [[[2.323983716480771, -4.427087042913458], [2.7666924207721166, -4.19468867126538], [2.5342940491240395, -3.751979966974035], [2.0915853448326938, -3.984378338622112], [2.323983716480771, -4.427087042913458]]]], "init": {"range": [[-1.0, 7.751070759394229], [-5.427087042913458, 3.323983716480771]], "scale": [30, 30]}, "line": [[[4.427087042913458, 2.323983716480771], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [2.323983716480771, -4.427087042913458], {"subtype": "segment"}], [[2.323983716480771, -4.427087042913458], [6.751070759394229, -2.103103326432687], {"subtype": "segment"}], [[6.751070759394229, -2.103103326432687], [4.427087042913458, 2.323983716480771], {"subtype": "segment"}]], "label": [[[1.9346654754790364, 1.6932423033900004], "5"], [[0.6307414130907705, -2.4924215674344214], "5"], [[4.816405283915192, -3.7963456298226874], "5"], [[6.120329346303458, 0.3893182410017343], "5"], [[4.427087042913458, 2.323983716480771], "H", "above", {}], [[0.0, 0.0], "I", "left", {}], [[2.323983716480771, -4.427087042913458], "J", "below", {}], [[6.751070759394229, -2.103103326432687], "K", "right", {}]]}
Exercice 3 : Démontrer qu'un quadrilatère est un losange
On considère le parallélogramme ci-dessous:
A l'aide des informations connues, démontrer que \(JKLM\) est un losange.
{"path": [[[[0.0, 0.0], [3.0, 0.0], [2.0, -1.0], [-1.0, -1.0], [0.0, 0.0]]], [[[1.0, -0.5], [0.8333333333333334, -0.4166666666666667], [1.01410871484888, -0.3714728212877799], [1.1807753815155468, -0.4548061546211133], [1.0, -0.5]]]], "init": {"range": [[-2.0, 4.0], [-2.0, 1.0]], "scale": [80, 80]}, "line": [[[0.0, 0.0], [1.0, -0.5], {"subtype": "segment", "stroke-dasharray": "- "}], [[3.0, 0.0], [1.0, -0.5], {"subtype": "segment", "stroke-dasharray": "- "}], [[1.0, -0.5], [2.0, -1.0], {"subtype": "segment", "stroke-dasharray": "- "}], [[1.0, -0.5], [-1.0, -1.0], {"subtype": "segment", "stroke-dasharray": "- "}]], "label": [[[0.0, 0.0], "J", "above left", {}], [[3.0, 0.0], "K", "right", {}], [[2.0, -1.0], "L", "below right", {}], [[-1.0, -1.0], "M", "left", {}], [[1.0, -0.5], "O", "below", {}]]}
A l'aide des informations connues, démontrer que \(JKLM\) est un losange.
Exercice 4 : Reconnaître un rectangle par ses côtés et ses angles
Parmi les quadrilatères suivants, lesquels sont des rectangles ?
- 1.
{"path": [[[[4.977937295262612, 0.4691910958591886], [0.0, 0.0], [0.8445439725465395, -8.960287131472702], [5.822481267809152, -8.491096035613515], [4.977937295262612, 0.4691910958591886]]], [[[0.0, 0.0], [0.4977937295262613, 0.04691910958591886], [0.5447128391121802, -0.45087461994034234], [0.046919109585918856, -0.4977937295262612], [0.0, 0.0]]], [[[4.977937295262612, 0.4691910958591886], [4.480143565736351, 0.42227198627326973], [4.52706267532227, -0.07552174325299155], [5.024856404848531, -0.028602633667072715], [4.977937295262612, 0.4691910958591886]]], [[[5.822481267809152, -8.491096035613515], [5.7755621582232335, -7.993302306087252], [5.277768428696972, -8.040221415673171], [5.324687538282891, -8.538015145199433], [5.822481267809152, -8.491096035613515]]], [[[0.8445439725465395, -8.960287131472702], [1.342337702072801, -8.913368021886784], [1.295418592486882, -8.415574292360523], [0.7976248629606206, -8.46249340194644], [0.8445439725465395, -8.960287131472702]]]], "init": {"range": [[-1.0, 6.822481267809152], [-9.960287131472702, 1.4691910958591887]], "scale": [30, 30]}, "line": [[[4.977937295262612, 0.4691910958591886], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [0.8445439725465395, -8.960287131472702], {"subtype": "segment"}], [[0.8445439725465395, -8.960287131472702], [5.822481267809152, -8.491096035613515], {"subtype": "segment"}], [[5.822481267809152, -8.491096035613515], [4.977937295262612, 0.4691910958591886], {"subtype": "segment"}]], "label": [[[2.4326657161282035, 0.8319480233611078], "5"], [[-0.17508048915824376, -4.536446497239454], "9"], [[3.3898155516809485, -9.323044058974622], "5"], [[5.997561756967396, -3.95464953837406], "9"], [[4.977937295262612, 0.4691910958591886], "M", "above right", {}], [[0.0, 0.0], "N", "above left", {}], [[0.8445439725465395, -8.960287131472702], "O", "below left", {}], [[5.822481267809152, -8.491096035613515], "P", "below right", {}]]}
- 2.
{"path": [[[[-7.31795701094465, -5.239036665835241], [0.0, 0.0], [-4.656921480742436, 6.504850676395245], [-13.629402018920503, -2.2123574771373886], [-7.31795701094465, -5.239036665835241]]], [[[0.0, 0.0], [-0.40655316727470275, -0.29105759254640223], [-0.697610759821105, 0.11549557472830058], [-0.29105759254640223, 0.4065531672747028], [0.0, 0.0]]]], "init": {"range": [[-14.629402018920503, 1.0], [-6.239036665835241, 7.504850676395245]], "scale": [30, 30]}, "line": [[[-7.31795701094465, -5.239036665835241], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-4.656921480742436, 6.504850676395245], {"subtype": "segment"}], [[-4.656921480742436, 6.504850676395245], [-13.629402018920503, -2.2123574771373886], {"subtype": "segment"}], [[-13.629402018920503, -2.2123574771373886], [-7.31795701094465, -5.239036665835241], {"subtype": "segment"}]], "label": [[[-3.3097093944166427, -3.107382133647264], "9"], [[-1.8405969396415744, 3.601694449253305], "8"], [[-9.561259913525063, 2.5765882377677976], "13"], [[-10.733122065942352, -4.266704985222833], "7"], [[-7.31795701094465, -5.239036665835241], "J", "below", {}], [[0.0, 0.0], "K", "right", {}], [[-4.656921480742436, 6.504850676395245], "L", "above", {}], [[-13.629402018920503, -2.2123574771373886], "M", "left", {}]]}
- 3.
{"path": [[[[-4.357489166888809, 2.4519967700726437], [0.0, 0.0], [2.9423961240871725, 5.22898700026657], [-1.4150930428016362, 7.680983770339214], [-4.357489166888809, 2.4519967700726437]]], [[[0.0, 0.0], [-0.4357489166888809, 0.2451996770072644], [-0.19054923968161652, 0.6809485936961452], [0.24519967700726436, 0.4357489166888808], [0.0, 0.0]]], [[[-4.357489166888809, 2.4519967700726437], [-3.9217402501999277, 2.2067970930653793], [-3.6765405731926633, 2.6425460097542603], [-4.112289489881544, 2.8877456867615248], [-4.357489166888809, 2.4519967700726437]]], [[[-1.4150930428016362, 7.680983770339214], [-1.6602927198089004, 7.2452348536503335], [-1.2245438031200195, 7.000035176643069], [-0.9793441261127552, 7.435784093331949], [-1.4150930428016362, 7.680983770339214]]], [[[2.9423961240871725, 5.22898700026657], [2.5066472073982915, 5.474186677273835], [2.261447530391027, 5.038437760584954], [2.697196447079908, 4.793238083577689], [2.9423961240871725, 5.22898700026657]]]], "init": {"range": [[-5.357489166888809, 3.9423961240871725], [-1.0, 8.680983770339214]], "scale": [30, 30]}, "line": [[[-1.9517617188790226, 1.2895147982635273], [-2.4057274480097863, 1.1624819718091164]], [[-2.24226099667161, 1.4529812496017036], [-2.1152281702171987, 0.9990155204709401]], [[-4.357489166888809, 2.4519967700726437], [0.0, 0.0], {"subtype": "segment"}], [[1.4076816488163808, 2.8414763646986665], [1.5347144752707917, 2.3875106355679034]], [[1.3259484231472927, 2.696226725802373], [1.4529812496017036, 2.24226099667161]], [[1.489414874485469, 2.98672600359496], [1.6164477009398799, 2.532760274464197]], [[0.0, 0.0], [2.9423961240871725, 5.22898700026657], {"subtype": "segment"}], [[0.5366686760773864, 6.391468972075687], [0.9906344052081499, 6.518501798530098]], [[0.8271679538699737, 6.228002520737511], [0.7001351274155627, 6.681968249868274]], [[2.9423961240871725, 5.22898700026657], [-1.4150930428016362, 7.680983770339214], {"subtype": "segment"}], [[-2.8227746916180165, 4.839507405640547], [-2.949807518072428, 5.29347313477131]], [[-2.7410414659489284, 4.984757044536841], [-2.8680742924033398, 5.438722773667604]], [[-2.9045079172871047, 4.694257766744253], [-3.031540743741516, 5.148223495875016]], [[-1.4150930428016362, 7.680983770339214], [-4.357489166888809, 2.4519967700726437], {"subtype": "segment"}]], "label": [[[-4.357489166888809, 2.4519967700726437], "C", "left", {}], [[0.0, 0.0], "D", "below", {}], [[2.9423961240871725, 5.22898700026657], "E", "right", {}], [[-1.4150930428016362, 7.680983770339214], "F", "above", {}]]}
- 4.
{"path": [[[[3.0244757410966265, -6.312887333979438], [0.0, 0.0], [-5.411046286268089, -2.5924077780828227], [-2.3865705451714625, -8.90529511206226], [3.0244757410966265, -6.312887333979438]]], [[[0.0, 0.0], [0.2160339815069019, -0.4509205238556741], [-0.23488654234877215, -0.6669545053625761], [-0.45092052385567405, -0.2160339815069019], [0.0, 0.0]]], [[[3.0244757410966265, -6.312887333979438], [2.8084417595897246, -5.861966810123763], [2.3575212357340507, -6.078000791630665], [2.5735552172409526, -6.52892131548634], [3.0244757410966265, -6.312887333979438]]], [[[-2.3865705451714625, -8.90529511206226], [-1.9356500213157886, -8.689261130555359], [-2.1516840028226905, -8.238340606699685], [-2.6026045266783644, -8.454374588206587], [-2.3865705451714625, -8.90529511206226]]], [[[-5.411046286268089, -2.5924077780828227], [-5.195012304761187, -3.043328301938497], [-4.744091780905514, -2.8272943204315952], [-4.9601257624124155, -2.376373796575921], [-5.411046286268089, -2.5924077780828227]]]], "init": {"range": [[-6.411046286268089, 4.0244757410966265], [-9.90529511206226, 1.0]], "scale": [30, 30]}, "line": [[[1.245456068403283, -3.06248905005021], [1.7790196726933436, -3.250398283929228]], [[3.0244757410966265, -6.312887333979438], [0.0, 0.0], {"subtype": "segment"}], [[-2.7994777600735534, -1.5629856911864417], [-2.6115685261945356, -1.029422086896381]], [[-2.6191095505312836, -1.4765720985836812], [-2.4312003166522658, -0.9430084942936202]], [[0.0, 0.0], [-5.411046286268089, -2.5924077780828227], {"subtype": "segment"}], [[-3.632026613574746, -5.84280606201205], [-4.165590217864807, -5.654896828133033]], [[-5.411046286268089, -2.5924077780828227], [-2.3865705451714625, -8.90529511206226], {"subtype": "segment"}], [[0.4129072149020908, -7.342309420875819], [0.22499798102307308, -7.8758730251658795]], [[0.2325390053598212, -7.42872301347858], [0.04462977148080344, -7.96228661776864]], [[-2.3865705451714625, -8.90529511206226], [3.0244757410966265, -6.312887333979438], {"subtype": "segment"}]], "label": [[[3.0244757410966265, -6.312887333979438], "W", "below right", {}], [[0.0, 0.0], "X", "above", {}], [[-5.411046286268089, -2.5924077780828227], "Y", "above left", {}], [[-2.3865705451714625, -8.90529511206226], "Z", "below", {}]]}
Exercice 5 : Reconnaître un losange par ses diagonales
Parmi les quadrilatères suivants, lesquels sont des losanges ?
- 1.
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- 2.
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- 3.
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- 4.
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