Le plan - 5e

Parallélogrammes & Obtention

Exercice 1 : Démontrer qu'un quadrilatère est un losange

On considère le quadrilatère ci-dessous:

A l'aide des informations connues, démontrer que \(QRST\) est un losange.

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210, "previous_statement": true, "next_statement": true, "compound": {"blocks": {"EqualityBetweenSides": {"inputs": [{"fields": [{"default": "", "type": "text_input", "name": "element"}, {"text": "a ses quatre c\u00f4t\u00e9s de m\u00eame longueur", "type": "text"}], "align": "left", "type": "dummy_input"}], "colour": 210, "next_statement": true, "previous_statement": true}}, "ast": {"type": "EqualityBetweenSides", "element": "QRST"}}}, "Start": {"inputs": [{"fields": [{"text": "D\u00e9monstration", "type": "text"}], "align": "left", "type": "dummy_input"}], "on_event": "start", "creatable": false, "tooltip": "D\u00e9marrage de la d\u00e9monstration", "colour": 65, "next_statement": true}, "VarValueLineBD": {"colour": 330, "output": "Line", "compound": {"blocks": {"VarValueLine": {"inputs": [{"fields": [{"default": "", "type": "text_input", "name": "value"}], "align": "left", "type": "dummy_input"}], "colour": 330, "output": "Line"}}, "ast": {"type": "VarValueLine", "value": "(RT)"}}}, 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"parallelogram", "type": "QuadrilateralProperty"}}}}, "ast": {"type": "Parallelogram", "element": "QRST"}}}, "SimpleComparison": {"inputs": [{"fields": [], "align": "left", "type": "value_input", "name": "value_1", "check": null}, {"fields": [{"type": "dropdown", "values": [["=", "EQ"], ["\u2260", "NE"], ["<", "LT"], ["\u2264", "LE"], [">", "GT"], ["\u2265", "GE"], ["appartient \u00e0", "BELONGS"]], "name": "op"}], "align": "left", "type": "dummy_input"}, {"fields": [], "align": "left", "type": "value_input", "name": "value_2", "check": null}], "colour": 210, "next_statement": true, "previous_statement": true}, "Implication": {"inputs": [{"fields": [{"text": "On sait que", "type": "text"}], "align": "left", "type": "dummy_input"}, {"fields": [], "align": "left", "type": "statement_input", "name": "hypo", "check": null}, {"fields": [{"text": "or,", "type": "text"}], "align": "left", "type": "dummy_input"}, {"fields": [], "align": "left", "type": "statement_input", "name": "theo", "check": null}, {"fields": [{"text": "donc", "type": "text"}], "align": "left", "type": "dummy_input"}, {"fields": [], "align": "left", "type": "statement_input", "name": "concl", "check": null}], "colour": 120, "next_statement": true, "tooltip": "\u00c0 partir d'hypoth\u00e8ses et d'une propri\u00e9t\u00e9 d\u00e9montre un r\u00e9sultat", "previous_statement": true}}}, "ast": [[{"editable": false, "type": "Start", "deletable": false, "movable": false}, {"theo": [], "editable": false, "concl": [{"editable": false, "type": "LosangeABCD", "deletable": false, "movable": false}], "movable": false, "hypo": [{"bool1": [{"bool1": [], "type": "And", "bool2": []}], "bool2": [], "editable": false, "movable": false, "deletable": false, "type": "And"}], "deletable": false, "type": "Implication"}]]}

Exercice 2 : Démontrer qu'un quadrilatère est un parallélogramme

Voici le quadrilatère \(ABCD\) :

On sait que \[(AB) // (CD)\] \[AB = CD = 3,4 mm\]
Compléter la démonstration pour montrer que le quadrilatère \(ABCD\) est un parallélogramme.

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coupent en", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "leur milieu alors c'est un", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "parall\u00e9logramme", "type": "text"}], "align": "left"}]}, "Parallel": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": "Line"}, {"type": "dummy_input", "fields": [{"text": "//", "type": "text"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": "Line"}]}, "theorem_Parallelogram3": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Si un quadrilat\u00e8re a deux", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "c\u00f4t\u00e9s oppos\u00e9s parall\u00e8les et de", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "m\u00eame longueur alors c'est un", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "parall\u00e9logramme", "type": "text"}], "align": "left"}]}, "VarValueLineAB": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(AB)"}}, "output": "Line", "colour": 330}, "VarValueLineCD": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(CD)"}}, "output": "Line", "colour": 330}, "Start": {"next_statement": true, "colour": 65, "tooltip": "D\u00e9marrage de la d\u00e9monstration", "inputs": [{"type": "dummy_input", "fields": [{"text": "D\u00e9monstration", 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Exercice 3 : Reconnaître un losange par ses côtés et ses angles

Parmi les quadrilatères suivants, lesquels sont des losanges ?

1.

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2.

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3.

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4.

{"path": [[[[-7.549168509342197, -2.6476508111071055], [0.0, 0.0], [-4.968108137211498, 6.270118941214186], [-14.040208811873097, -0.027230846498463565], [-7.549168509342197, -2.6476508111071055]]]], "init": {"range": [[-15.040208811873097, 1.0], [-3.6476508111071055, 7.270118941214186]], "scale": [30, 30]}, "line": [[[-7.549168509342197, -2.6476508111071055], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-4.968108137211498, 6.270118941214186], {"subtype": "segment"}], [[-4.968108137211498, 6.270118941214186], [-14.040208811873097, -0.027230846498463565], {"subtype": "segment"}], [[-14.040208811873097, -0.027230846498463565], [-7.549168509342197, -2.6476508111071055], {"subtype": "segment"}]], "label": [[[-3.5760104438380655, -1.8900130437542175], "8"], [[-2.0137822521028994, 3.5076777989312777], "8"], [[-9.846296249886215, 3.6143352551134007], "11"], [[-11.019295616304941, -1.893814548035095], "7"], [[-7.549168509342197, -2.6476508111071055], "H", "below", {}], [[0.0, 0.0], "I", "right", {}], [[-4.968108137211498, 6.270118941214186], "J", "above", {}], [[-14.040208811873097, -0.027230846498463565], "K", "left", {}]]}

Exercice 4 : Reconnaître un parallélogramme par ses diagonales

Parmi les quadrilatères suivants, lesquels sont des parallélogrammes ?

1.

{"init": {"range": [[-11.722782723799964, 1.0], [-5.656495815892331, 1.922514072742397]], "scale": [30, 30]}, "path": [[[[-3.783787350866485, -4.656495815892331], [0.0, 0.0], [-6.938995372933479, 0.9225140727423972], [-10.722782723799964, -3.733981743149934], [-3.783787350866485, -4.656495815892331]], {"names": ["S", "T", "U", "V"]}]], "line": [[[-2.8204717005629916, -1.2226307251264814], [-2.5409196613369907, -0.6443601464484856]], [[-2.6060160460869923, -1.1479510902634829], [-2.3264640068609914, -0.5696805115854868]], [[0.0, 0.0], [-5.361391361899982, -1.866990871574967], {"subtype": "segment"}], [[-7.902311023236972, -2.5113510180234524], [-8.181863062462973, -3.0896215967014484]], [[-8.116766677712972, -2.586030652886451], [-8.396318716938973, -3.164301231564447]], [[-10.722782723799964, -3.733981743149934], [-5.361391361899982, -1.866990871574967], {"subtype": "segment"}], [[-3.783787350866485, -4.656495815892331], [-6.938995372933479, 0.9225140727423972], {"subtype": "segment"}], [[0.0, 0.0], [-10.722782723799964, -3.733981743149934], {"subtype": "segment"}]], "label": [[[-3.783787350866485, -4.656495815892331], "S", "below right", {}], [[0.0, 0.0], "T", "right", {}], [[-6.938995372933479, 0.9225140727423972], "U", "above left", {}], [[-10.722782723799964, -3.733981743149934], "V", "left", {}], [[-5.361391361899982, -1.866990871574967], "O", "left", {}]]}
Sachant que \(SO = OU\)
2.

{"init": {"range": [[-2.04860264957492, 7.018437005266064], [-1.0, 12.1786045406836]], "scale": [30, 30]}, "path": [[[[-1.0486026495749199, 5.907658798822462], [0.0, 0.0], [6.018437005266064, 5.270945741861137], [4.969834355691144, 11.1786045406836], [-1.0486026495749199, 5.907658798822462]], {"names": ["C", "D", "E", "F"]}]], "label": [[[-1.0486026495749199, 5.907658798822462], "C", "left", {}], [[0.0, 0.0], "D", "below left", {}], [[6.018437005266064, 5.270945741861137], "E", "right", {}], [[4.969834355691144, 11.1786045406836], "F", "above right", {}], [[2.484917177845572, 5.5893022703418], "O", "above right", {}]], "line": [[[-1.0486026495749199, 5.907658798822462], [6.018437005266064, 5.270945741861137], {"subtype": "segment"}], [[0.0, 0.0], [4.969834355691144, 11.1786045406836], {"subtype": "segment"}]]}
Sachant que \(O\) est le milieu de \([CE]\)
3.

{"init": {"range": [[-2.282894429584074, 5.826990079527422], [-8.165161200144254, 1.0]], "scale": [30, 30]}, "path": [[[[4.826990079527422, -1.3039044336698349], [0.0, 0.0], [-1.2828944295840738, -5.861256766474419], [3.544095649943348, -7.1651612001442535], [4.826990079527422, -1.3039044336698349]], {"names": ["D", "E", "F", "G"]}]], "label": [[[0.3482171231969492, -2.057305069165106], "4"], [[2.1202649481686233, -5.639885669237233], "4"], [[4.826990079527422, -1.3039044336698349], "D", "above right", {}], [[0.0, 0.0], "E", "above left", {}], [[-1.2828944295840738, -5.861256766474419], "F", "below left", {}], [[3.544095649943348, -7.1651612001442535], "G", "below right", {}], [[1.7720478249716742, -3.582580600072127], "O", "below", {}]], "line": [[[0.0, 0.0], [1.7720478249716742, -3.582580600072127], {"subtype": "segment"}], [[1.7720478249716742, -3.582580600072127], [3.544095649943348, -7.1651612001442535], {"subtype": "segment"}], [[4.826990079527422, -1.3039044336698349], [-1.2828944295840738, -5.861256766474419], {"subtype": "segment"}], [[0.0, 0.0], [3.544095649943348, -7.1651612001442535], {"subtype": "segment"}]]}
Sachant que \(O\) est le milieu de \([DF]\)
4.

{"init": {"range": [[-1.0, 11.760562325517032], [-1.7421882099392705, 4.249124874209729]], "scale": [30, 30]}, "path": [[[[3.8004193915660958, 3.2491248742097287], [0.0, 0.0], [6.9601429339509355, -0.7421882099392706], [10.760562325517032, 2.5069366642704582], [3.8004193915660958, 3.2491248742097287]], {"names": ["I", "J", "K", "L"]}]], "line": [[[2.8552130946041894, 0.8920841458633644], [2.5250680681543267, 0.36138418627186475]], [[2.6400018480938487, 0.8419454125779552], [2.309856821643986, 0.31124545298645556]], [[3.07042434111453, 0.9422228791487736], [2.7402793146646673, 0.41152291955727394]], [[0.0, 0.0], [5.380281162758516, 1.2534683321352291], {"subtype": "segment"}], [[7.905349230912844, 1.6148525184070937], [8.235494257362706, 2.1455524779985935]], [[8.120560477423185, 1.664991251692503], [8.450705503873047, 2.1956912112840024]], [[7.690137984402503, 1.5647137851216848], [8.020283010852365, 2.095413744713184]], [[10.760562325517032, 2.5069366642704582], [5.380281162758516, 1.2534683321352291], {"subtype": "segment"}], [[3.8004193915660958, 3.2491248742097287], [6.9601429339509355, -0.7421882099392706], {"subtype": "segment"}], [[0.0, 0.0], [10.760562325517032, 2.5069366642704582], {"subtype": "segment"}]], "label": [[[3.8004193915660958, 3.2491248742097287], "I", "above left", {}], [[0.0, 0.0], "J", "left", {}], [[6.9601429339509355, -0.7421882099392706], "K", "below right", {}], [[10.760562325517032, 2.5069366642704582], "L", "right", {}], [[5.380281162758516, 1.2534683321352291], "O", "right", {}]]}

Exercice 5 : Reconnaître un quadrilatère grâce à un cercle

Dans la figure ci-dessous, les deux cercles sont concentriques. Donner la nature du quadrilatère \(MLKJ\).

Revenir au choix du niveau