Quadrilatères particuliers

Figures planes - Mathématiques 6e

Exercice 1 : Reconnaître un carré par ses diagonales

Parmi les quadrilatères suivants, lesquels sont des carrés ?

1.

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2.

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3.

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4.

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Exercice 2 : Reconnaître un rectangle par ses diagonales

Parmi les quadrilatères suivants, lesquels sont des rectangles ?

1.

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2.

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3.

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4.

{"path": [[[[-3.9851352337995043, 4.485387070067618], [0.0, 0.0], [5.980516093423491, 5.313513645066005], [1.9953808596239868, 9.798900715133623], [-3.9851352337995043, 4.485387070067618]]]], "init": {"range": [[-4.985135233799504, 6.980516093423491], [-1.0, 10.798900715133623]], "scale": [30, 30]}, "line": [[[-1.6764728970382476, 4.476543155772787], [-1.3109719069492631, 4.908294271861643]], [[0.9976904298119933, 4.899450357566812], [-3.9851352337995043, 4.485387070067618], {"subtype": "segment", "stroke-dasharray": "-"}], [[0.6549156120161894, 2.2138395472882535], [0.3427748177958039, 2.685610810278558]], [[0.9976904298119933, 4.899450357566812], [0.0, 0.0], {"subtype": "segment", "stroke-dasharray": "-"}], [[3.6718537566622342, 5.322357559360837], [3.30635276657325, 4.890606443271981]], [[0.9976904298119933, 4.899450357566812], [5.980516093423491, 5.313513645066005], {"subtype": "segment", "stroke-dasharray": "-"}], [[1.3404652476077974, 7.585061167845369], [1.6526060418281827, 7.1132899048550655]], [[0.9976904298119933, 4.899450357566812], [1.9953808596239868, 9.798900715133623], {"subtype": "segment", "stroke-dasharray": "-"}]], "label": [[[-3.9851352337995043, 4.485387070067618], "E", "left", {}], [[0.0, 0.0], "F", "below", {}], [[5.980516093423491, 5.313513645066005], "G", "right", {}], [[1.9953808596239868, 9.798900715133623], "H", "above", {}]]}

Exercice 3 : Problème - Propriétés des diagonales et triangles particuliers dans un rectangle/carré/losange

\( RSTU \) est un losange dont les diagonales se coupent en \( O \).
On sait que \( 14\:\text{cm} = RT \).

Donner la longueur de \( [SU] \) si l'énoncé le permet. Dans le cas contraire, écrire "Impossible".

Donner la longueur de \( [SO] \) si l'énoncé le permet. Dans le cas contraire, écrire "Impossible".

Quelle est la nature du triangle \( RST \) ?
On choisira la réponse la plus précise possible.

Quelle est la nature du triangle \( ROS \) ?
On choisira la réponse la plus précise possible.

C'est un triangle rectangle

C'est un triangle isocèle

C'est un triangle rectangle isocèle

C'est un triangle quelconque

Exercice 4 : Reconnaître un losange par ses côtés et ses angles

Parmi les quadrilatères suivants, lesquels sont des losanges ?

1.

{"path": [[[[-7.129937119823884, -3.6282222461361844], [0.0, 0.0], [-4.705581463477287, 1.6907729269712573], [-16.077400403946967, -4.599813610480953], [-7.129937119823884, -3.6282222461361844]]]], "init": {"range": [[-17.077400403946967, 1.0], [-5.599813610480953, 2.6907729269712575]], "scale": [30, 30]}, "line": [[[-7.129937119823884, -3.6282222461361844], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-4.705581463477287, 1.6907729269712573], {"subtype": "segment"}], [[-4.705581463477287, 1.6907729269712573], [-16.077400403946967, -4.599813610480953], {"subtype": "segment"}], [[-16.077400403946967, -4.599813610480953], [-7.129937119823884, -3.6282222461361844], {"subtype": "segment"}]], "label": [[[-3.292851891451728, -2.3488564070548836], "8"], [[-2.1499028903284465, 1.4100425745903529], "5"], [[-10.681920450028478, -0.9294958698944877], "13"], [[-11.538896439674922, -4.710511470834771], "9"], [[-7.129937119823884, -3.6282222461361844], "N", "below", {}], [[0.0, 0.0], "O", "right", {}], [[-4.705581463477287, 1.6907729269712573], "P", "above right", {}], [[-16.077400403946967, -4.599813610480953], "Q", "left", {}]]}
2.

{"path": [[[[0.02824640374795271, 6.999943009816245], [0.0, 0.0], [9.00015673333712, 0.020682731118278796], [9.028403137085071, 7.020625740934524], [0.02824640374795271, 6.999943009816245]]]], "init": {"range": [[-1.0, 10.028403137085071], [-1.0, 8.020625740934523]], "scale": [30, 30]}, "line": [[[0.24651308874291944, 3.2656985244559826], [-0.2182666849949667, 3.734244485360262]], [[0.24839618232611627, 3.7323613917770655], [-0.22014977857816354, 3.2675816180391792]], [[0.02824640374795271, 6.999943009816245], [0.0, 0.0], {"subtype": "segment"}], [[4.732874875687361, 0.24421029098008587], [4.267281857649759, -0.22352755986180708]], [[4.499542158467487, 0.24367408277901306], [4.033949140429885, -0.2240637680628799]], [[4.966207592907234, 0.24474649918115868], [4.5006145748696325, -0.22299135166073428]], [[0.0, 0.0], [9.00015673333712, 0.020682731118278796], {"subtype": "segment"}], [[8.781890048342154, 3.754927216478541], [9.246669822080039, 3.2863812555742618]], [[8.780006954758957, 3.2882643491574584], [9.248552915663236, 3.7530441228953446]], [[9.00015673333712, 0.020682731118278796], [9.028403137085071, 7.020625740934524], {"subtype": "segment"}], [[4.2955282613977115, 6.776415449954438], [4.761121279435313, 7.244153300796331]], [[4.528860978617585, 6.776951658155511], [4.994453996655187, 7.2446895089974035]], [[4.062195544177838, 6.7758792417533655], [4.52778856221544, 7.243617092595258]], [[9.028403137085071, 7.020625740934524], [0.02824640374795271, 6.999943009816245], {"subtype": "segment"}]], "label": [[[0.02824640374795271, 6.999943009816245], "K", "above left", {}], [[0.0, 0.0], "L", "below left", {}], [[9.00015673333712, 0.020682731118278796], "M", "below right", {}], [[9.028403137085071, 7.020625740934524], "N", "above right", {}]]}
3.

{"path": [[[[-5.088043501281295, -3.1799077548050336], [0.0, 0.0], [-4.931052440420335, 3.4181524000290935], [-10.01909594170163, 0.23824464522405966], [-5.088043501281295, -3.1799077548050336]]]], "init": {"range": [[-11.01909594170163, 1.0], [-4.179907754805034, 4.4181524000290935]], "scale": [30, 30]}, "line": [[[-5.088043501281295, -3.1799077548050336], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-4.931052440420335, 3.4181524000290935], {"subtype": "segment"}], [[-4.931052440420335, 3.4181524000290935], [-10.01909594170163, 0.23824464522405966], {"subtype": "segment"}], [[-10.01909594170163, 0.23824464522405966], [-5.088043501281295, -3.1799077548050336], {"subtype": "segment"}]], "label": [[[-2.226030975160144, -2.098758227530646], "6"], [[-2.1237064415700657, 2.202187991528836], "6"], [[-7.793064966541485, 2.337002872754706], "6"], [[-7.895389500131564, -1.963943346304776], "6"], [[-5.088043501281295, -3.1799077548050336], "L", "below", {}], [[0.0, 0.0], "M", "right", {}], [[-4.931052440420335, 3.4181524000290935], "N", "above", {}], [[-10.01909594170163, 0.23824464522405966], "O", "left", {}]]}
4.

{"path": [[[[-4.440210166541915, 2.298811361755858], [0.0, 0.0], [0.7655102539176077, 4.941260775464801], [-3.674699912624307, 7.2400721372206585], [-4.440210166541915, 2.298811361755858]]]], "init": {"range": [[-5.440210166541915, 1.7655102539176077], [-1.0, 8.24007213722066]], "scale": [30, 30]}, "line": [[[-1.9954710323276983, 1.2207856410374642], [-2.4447391342142164, 1.0780257207183936]], [[-2.2914850434304928, 1.374039731821188], [-2.148725123111422, 0.9247716299346699]], [[-4.440210166541915, 2.298811361755858], [0.0, 0.0], {"subtype": "segment"}], [[0.24356918838419095, 2.6608482366810717], [0.5219410655334168, 2.280412538783729]], [[0.19253727801013232, 2.3314444491577873], [0.5729729759074754, 2.6098163263070133]], [[0.0, 0.0], [0.7655102539176077, 4.941260775464801], {"subtype": "segment"}], [[-1.6792288802966089, 6.019286496183194], [-1.2299607784100905, 6.162046416502265]], [[-1.3832148691938144, 5.86603240539947], [-1.525974789512885, 6.315300507285989]], [[0.7655102539176077, 4.941260775464801], [-3.674699912624307, 7.2400721372206585], {"subtype": "segment"}], [[-3.918269101008498, 4.579223900539587], [-4.196640978157724, 4.9596595984369305]], [[-3.8672371906344396, 4.908627688062872], [-4.247672888531783, 4.630255810913646]], [[-3.674699912624307, 7.2400721372206585], [-4.440210166541915, 2.298811361755858], {"subtype": "segment"}]], "label": [[[-4.440210166541915, 2.298811361755858], "F", "below left", {}], [[0.0, 0.0], "G", "below right", {}], [[0.7655102539176077, 4.941260775464801], "H", "above right", {}], [[-3.674699912624307, 7.2400721372206585], "I", "above left", {}]]}

Exercice 5 : Reconnaître des quadrilatères particuliers dans une figure complexe

On considère la figure ci-dessous :

{"init": {"range": [[-3.0, 4.0], [-4.0, 5.5]], "scale": [60, 60]}, "label": [[[2.0, 0.0], "V", "below right", {"color": "blue"}], [[2.0, 4.0], "D", "above right", {"color": "blue"}], [[2.0, 4.5], "K", "above right", {"color": "blue"}], [[-2.0, 4.5], "C", "above left", {"color": "blue"}], [[-2.0, 4.0], "I", "above left", {"color": "blue"}], [[-2.0, 0.0], "Q", "below left", {"color": "blue"}], [[0.0, -3.0], "S", "below", {"color": "blue"}], [[3.0, 4.0], "N", "above right", {"color": "blue"}], [[3.0, 0.0], "R", "below right", {"color": "blue"}], [[0.0, 3.0], "B", "above left", {"color": "blue"}]], "line": [[[1.9, 0.0], [2.1, 0.0], {"stroke": "blue"}], [[2.0, -0.1], [2.0, 0.1], {"stroke": "blue"}], [[1.9, 4.0], [2.1, 4.0], {"stroke": "blue"}], [[2.0, 3.9], [2.0, 4.1], {"stroke": "blue"}], [[1.9, 4.5], [2.1, 4.5], {"stroke": "blue"}], [[2.0, 4.4], [2.0, 4.6], {"stroke": "blue"}], [[-2.1, 4.5], [-1.9, 4.5], {"stroke": "blue"}], [[-2.0, 4.4], [-2.0, 4.6], {"stroke": "blue"}], [[-2.1, 4.0], [-1.9, 4.0], {"stroke": "blue"}], [[-2.0, 3.9], [-2.0, 4.1], {"stroke": "blue"}], [[-2.1, 0.0], [-1.9, 0.0], {"stroke": "blue"}], [[-2.0, -0.1], [-2.0, 0.1], {"stroke": "blue"}], [[-0.1, -3.0], [0.1, -3.0], {"stroke": "blue"}], [[0.0, -3.1], [0.0, -2.9], {"stroke": "blue"}], [[2.9, 4.0], [3.1, 4.0], {"stroke": "blue"}], [[3.0, 3.9], [3.0, 4.1], {"stroke": "blue"}], [[2.9, 0.0], [3.1, 0.0], {"stroke": "blue"}], [[3.0, -0.1], [3.0, 0.1], {"stroke": "blue"}], [[-0.1, 3.0], [0.1, 3.0], {"stroke": "blue"}], [[0.0, 2.9], [0.0, 3.1], {"stroke": "blue"}], [[2.16, 1.84], [1.84, 2.16]], [[2.0, 0.0], [2.0, 4.0], {"subtype": "segment"}], [[0.16, 4.16], [-0.16, 3.84]], [[2.0, 4.0], [-2.0, 4.0], {"subtype": "segment"}], [[0.16, 0.16], [-0.16, -0.16]], [[2.0, 0.0], [-2.0, 0.0], {"subtype": "segment"}], [[-1.84, 1.84], [-2.16, 2.16]], [[-2.0, 4.0], [-2.0, 0.0], {"subtype": "segment"}], [[0.16, 4.66], [-0.16, 4.34]], [[-2.0, 4.5], [2.0, 4.5], {"subtype": "segment"}], [[0.8, 1.54], [1.2, 1.46]], [[0.8800000000000001, 1.42], [1.28, 1.3399999999999999]], [[2.0, 0.0], [0.0, 3.0], {"subtype": "segment"}], [[-0.96, 1.3], [-1.04, 1.7]], [[-0.88, 1.4200000000000002], [-0.9600000000000001, 1.82]], [[0.0, 3.0], [-2.0, 0.0], {"subtype": "segment"}], [[-0.8, -1.54], [-1.2, -1.46]], [[-0.8800000000000001, -1.42], [-1.28, -1.3399999999999999]], [[-2.0, 0.0], [0.0, -3.0], {"subtype": "segment"}], [[0.96, -1.3], [1.04, -1.7]], [[0.88, -1.4200000000000002], [0.9600000000000001, -1.82]], [[0.0, -3.0], [2.0, 0.0], {"subtype": "segment"}], [[2.0, 4.0], [2.0, 4.5], {"subtype": "segment"}], [[-2.0, 4.5], [-2.0, 4.0], {"subtype": "segment"}], [[3.16, 1.84], [2.84, 2.16]], [[3.0, 4.0], [3.0, 0.0], {"subtype": "segment"}], [[2.0, 4.0], [3.0, 4.0], {"subtype": "segment"}], [[2.0, 0.0], [3.0, 0.0], {"subtype": "segment"}]], "path": [[[[-2.0, 4.5], [-2.0, 4.3], [-1.8, 4.3], [-1.8, 4.5], [-2.0, 4.5]], {"stroke": "red"}], [[[2.0, 4.5], [1.8, 4.5], [1.8, 4.3], [2.0, 4.3], [2.0, 4.5]], {"stroke": "red"}], [[[-2.0, 4.0], [-2.0, 3.8], [-1.8, 3.8], [-1.8, 4.0], [-2.0, 4.0]], {"stroke": "red"}], [[[2.0, 4.0], [2.0, 3.8], [1.8, 3.8], [1.8, 4.0], [2.0, 4.0]], {"stroke": "red"}], [[[-2.0, 0.0], [-2.0, 0.2], [-1.8, 0.2], [-1.8, 0.0], [-2.0, 0.0]], {"stroke": "red"}], [[[2.0, 0.0], [1.8, 0.0], [1.8, 0.2], [2.0, 0.2], [2.0, 0.0]], {"stroke": "red"}], [[[3.0, 4.0], [2.8, 4.0], [2.8, 3.8], [3.0, 3.8], [3.0, 4.0]], {"stroke": "red"}], [[[3.0, 0.0], [3.0, 0.2], [2.8, 0.2], [2.8, 0.0], [3.0, 0.0]], {"stroke": "red"}]]}

Combien y-a-t'il de quadrilatères sur cette figure ?

Donner la nature exacte du quadrilatère \(VKCQ\).

C'est un carré

C'est un losange

C'est un quadrilatère quelconque

C'est un rectangle

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