Droites perpendiculaires

Parallélisme et perpendicularité - Mathématiques 6e

Exercice 1 : Problème - Droites perpendiculaires, parallèles, angles droits et nature de quadrilatère

On s'intéresse à la figure ci-dessous :

En plus des marquages, on sait que les droites \( (AB) \) et \( (DC) \) sont parallèles.

En utilisant le codage ou en raisonnant, répondre aux questions suivantes.

Parmi ces droites, lesquelles sont parallèles entre elles ?

A. \( (AE) \) et \( (FB) \)
B. \( (DA) \) et \( (CB) \)
C. \( (AB) \) et \( (DC) \)

Parmi ces angles, le ou lesquels sont des angles droits ?

A. \( \widehat{EFB} \)
B. \( \widehat{AEF} \)
C. \( \widehat{BCD} \)

Quelle est la nature du quadrilatère \( ABCD \) ?

Exercice 2 : Tracer sur quadrillage la parallèle et la perpendiculaire à une droite (parmi plusieurs) passant par un point

Construire la parallèle et la perpendiculaire à \((r)\) passant par \(N\).

{"init": {"range": [[-9, 9], [-9, 9]], "scale": [35, 35]}, "line": [[[-9.0, -9.0], [9.0, -9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -8.0], [9.0, -8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -7.0], [9.0, -7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -6.0], [9.0, -6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -5.0], [9.0, -5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -4.0], [9.0, -4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -3.0], [9.0, -3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -2.0], [9.0, -2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -1.0], [9.0, -1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 0.0], [9.0, 0.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 1.0], [9.0, 1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 2.0], [9.0, 2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 3.0], [9.0, 3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 4.0], [9.0, 4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 5.0], [9.0, 5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 6.0], [9.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 7.0], [9.0, 7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 8.0], [9.0, 8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -9.0], [-9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-8.0, -9.0], [-8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-7.0, -9.0], [-7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -9.0], [-6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-5.0, -9.0], [-5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-4.0, -9.0], [-4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-3.0, -9.0], [-3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.0, -9.0], [-2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.0, -9.0], [-1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[0.0, -9.0], [0.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[1.0, -9.0], [1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[2.0, -9.0], [2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.0, -9.0], [3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[4.0, -9.0], [4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[5.0, -9.0], [5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[6.0, -9.0], [6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[7.0, -9.0], [7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[8.0, -9.0], [8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[9.0, -9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.9, 0.0], [4.1, 0.0], {}], [[4.0, -0.1], [4.0, 0.1], {}], [[9.0, 0.0], [0.0, -9.0], {"subtype": "line"}], [[-5.0, 9.0], [-9.0, -3.0], {"subtype": "line"}]], "label": [[[4.0, 0.0], "N", "above", {}], [[1.0, -8.0], "(w)", "above left", {"color": "blue"}], [[-7.0, 3.0], "(r)", "above left", {"color": "blue"}]], "drawing_options": {"tools": ["line"]}}

Essais restants : 2

Exercice 3 : Tracer sur quadrillage les parallèles ou les perpendiculaires à deux droites passant par un point

Construire les parallèles passant par \(H\) de chacune de ces deux droites.

{"init": {"range": [[-9, 9], [-9, 9]], "scale": [35, 35]}, "line": [[[-9.0, -9.0], [9.0, -9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -8.0], [9.0, -8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -7.0], [9.0, -7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -6.0], [9.0, -6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -5.0], [9.0, -5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -4.0], [9.0, -4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -3.0], [9.0, -3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -2.0], [9.0, -2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -1.0], [9.0, -1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 0.0], [9.0, 0.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 1.0], [9.0, 1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 2.0], [9.0, 2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 3.0], [9.0, 3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 4.0], [9.0, 4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 5.0], [9.0, 5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 6.0], [9.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 7.0], [9.0, 7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 8.0], [9.0, 8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -9.0], [-9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-8.0, -9.0], [-8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-7.0, -9.0], [-7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -9.0], [-6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-5.0, -9.0], [-5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-4.0, -9.0], [-4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-3.0, -9.0], [-3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.0, -9.0], [-2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.0, -9.0], [-1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[0.0, -9.0], [0.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[1.0, -9.0], [1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[2.0, -9.0], [2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.0, -9.0], [3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[4.0, -9.0], [4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[5.0, -9.0], [5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[6.0, -9.0], [6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[7.0, -9.0], [7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[8.0, -9.0], [8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[9.0, -9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.1, 5.0], [-1.9, 5.0], {}], [[-2.0, 4.9], [-2.0, 5.1], {}], [[7.5, -9.0], [-9.0, 2.0], {"subtype": "line"}], [[-0.33333, 9.0], [-6.33333, -9.0], {"subtype": "line"}]], "label": [[[-2.0, 5.0], "H", "above", {}]], "drawing_options": {"tools": ["line"]}}

Essais restants : 2

Exercice 4 : Problème - Reconnaître des droites parallèles et des droites perpendiculaires

On considère l'ensemble de droites suivant :

{"init": {"range": [[-8, 8], [-3, 8]], "scale": [30, 30]}, "path": [[[[-1.5678664394403006, 3.679917774635297], [-1.1078767176108886, 3.8759010795653346], [-0.9118934126808509, 3.415911357735922], [-1.371883134510263, 3.219928052805885], [-1.5678664394403006, 3.679917774635297]]], [[[0.0, 0.0], [-0.19598330493003757, 0.4599897218294121], [0.26400641689937454, 0.6559730267594497], [0.4599897218294121, 0.19598330493003757], [0.0, 0.0]]], [[[3.679917774635297, 1.5678664394403006], [3.219928052805885, 1.371883134510263], [3.023944747875847, 1.8318728563396751], [3.483934469705259, 2.0278561612697126], [3.679917774635297, 1.5678664394403006]]], [[[2.1120513351949963, 5.2477842140755975], [2.308034640125034, 4.787794492246185], [1.8480449182956218, 4.591811187316148], [1.6520616133655843, 5.05180090914556], [2.1120513351949963, 5.2477842140755975]]], [[[-1.0, 5.0], [-0.5, 5.0], [-0.5, 4.5], [-1.0, 4.5], [-1.0, 5.0]]], [[[-1.0, 0.0], [-1.0, 0.5], [-0.5, 0.5], [-0.5, 0.0], [-1.0, 0.0]]], [[[4.0, 0.0], [3.5, 0.0], [3.5, 0.5], [4.0, 0.5], [4.0, 0.0]]], [[[4.0, 5.0], [4.0, 4.5], [3.5, 4.5], [3.5, 5.0], [4.0, 5.0]]]], "label": [[[-3.40848, 8.0], "(d1)", {"subtype": "line", "stroke": "#6495ed", "color": "#6495ed"}], [[8.0, 7.7564], "(d4)", {"subtype": "line", "stroke": "#b787df", "color": "#b787df"}], [[5.6261, -3.0], "(d3)", {"subtype": "line", "stroke": "#61175e", "color": "#61175e"}], [[-7.04126, -3.0], "(d7)", {"subtype": "line", "stroke": "#a37351", "color": "#a37351"}], [[-1.0, -3.0], "(d6)", {"subtype": "line", "stroke": "#ff6666", "color": "#ff6666"}], [[-8.0, 5.0], "(d2)", {"subtype": "line", "stroke": "#ffa400", "color": "#ffa400"}], [[4.0, 8.0], "(d5)", {"subtype": "line", "stroke": "#800000", "color": "#800000"}], [[8.0, 0.0], "(d8)", {"subtype": "line", "stroke": "#008000", "color": "#008000"}]], "line": [[[-3.2124967662045085, 7.5400102478629965], [1.27818, -3.0], {"subtype": "line", "stroke": "#6495ed", "color": "#6495ed"}], [[7.5400102478629965, 7.560416766204509], [-8.0, 0.93944], {"subtype": "line", "stroke": "#b787df", "color": "#b787df"}], [[5.430116766204509, -2.540010247862997], [0.93944, 8.0], {"subtype": "line", "stroke": "#61175e", "color": "#61175e"}], [[8.0, 3.40848], [-6.581270256292689, -2.804016746419287], {"subtype": "line", "stroke": "#a37351", "color": "#a37351"}], [[-1.0, 8.0], [-1.0, -2.5], {"subtype": "line", "stroke": "#ff6666", "color": "#ff6666"}], [[-7.5, 5.0], [8.0, 5.0], {"subtype": "line", "stroke": "#ffa400", "color": "#ffa400"}], [[4.0, -3.0], [4.0, 7.5], {"subtype": "line", "stroke": "#800000", "color": "#800000"}], [[7.5, 0.0], [-8.0, 0.0], {"subtype": "line", "stroke": "#008000", "color": "#008000"}]]}

Pour les deux questions suivantes, on donnera la réponse sous forme de liste séparée par des points-virgules.
S'il n'y en a pas, écrire "Aucune".

Donner la ou les droites parallèles à la droite \((d1)\).

Donner la ou les droites perpendiculaires à la droite \((d1)\).

Exercice 5 : Tracer sur quadrillage la parallèle et la perpendiculaire à une droite passant par un point

Construire la parallèle et la perpendiculaire à (d) passant par A.

{"init": {"range": [[-9, 9], [-9, 9]], "scale": [35, 35]}, "line": [[[-9.0, -9.0], [9.0, -9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -8.0], [9.0, -8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -7.0], [9.0, -7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -6.0], [9.0, -6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -5.0], [9.0, -5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -4.0], [9.0, -4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -3.0], [9.0, -3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -2.0], [9.0, -2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -1.0], [9.0, -1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 0.0], [9.0, 0.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 1.0], [9.0, 1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 2.0], [9.0, 2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 3.0], [9.0, 3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 4.0], [9.0, 4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 5.0], [9.0, 5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 6.0], [9.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 7.0], [9.0, 7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 8.0], [9.0, 8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, 9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -9.0], [-9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-8.0, -9.0], [-8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-7.0, -9.0], [-7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -9.0], [-6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-5.0, -9.0], [-5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-4.0, -9.0], [-4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-3.0, -9.0], [-3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.0, -9.0], [-2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.0, -9.0], [-1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[0.0, -9.0], [0.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[1.0, -9.0], [1.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[2.0, -9.0], [2.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.0, -9.0], [3.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[4.0, -9.0], [4.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[5.0, -9.0], [5.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[6.0, -9.0], [6.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[7.0, -9.0], [7.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[8.0, -9.0], [8.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[9.0, -9.0], [9.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.9, -3.0], [4.1, -3.0], {}], [[4.0, -3.1], [4.0, -2.9], {}], [[-9.0, -1.0], [9.0, -1.0], {"subtype": "line"}]], "label": [[[4.0, -3.0], "A", "above", {}], [[3.0, -1.0], "(d)", "above right", {}]], "drawing_options": {"tools": ["line"]}}

Essais restants : 2

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