Directions

Se repérer et se déplacer dans l'espace - Mathématiques 6e

Exercice 1 : Tracer un itinéraire à partir d'une séquence de directions (avec diagonales)

À partir du point \(D\) tracer l'itinéraire correspondant aux instructions suivantes : \[SO6 - E6 - NO3 - E1 - S3 - NO3 - E3\] Les chiffres indiquent une distance en nombre de carreaux. On considère qu'une diagonale compte pour \(\textit{1,5}\) coté de carreau.
Les lettres indiquent une direction (voir la rose des vents).

{"init": {"range": [[-7.0, 13.0], [-7.0, 7.0]], "scale": [35, 35]}, "line": [[[-6.0, -6.0], [12.0, -6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -5.0], [12.0, -5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -4.0], [12.0, -4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -3.0], [12.0, -3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -2.0], [12.0, -2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -1.0], [12.0, -1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 0.0], [12.0, 0.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 1.0], [12.0, 1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 2.0], [12.0, 2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 3.0], [12.0, 3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 4.0], [12.0, 4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 5.0], [12.0, 5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 6.0], [12.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -6.0], [-6.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-5.0, -6.0], [-5.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-4.0, -6.0], [-4.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-3.0, -6.0], [-3.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.0, -6.0], [-2.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.0, -6.0], [-1.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[0.0, -6.0], [0.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[1.0, -6.0], [1.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[2.0, -6.0], [2.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.0, -6.0], [3.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[4.0, -6.0], [4.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[5.0, -6.0], [5.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[6.0, -6.0], [6.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[7.0, -6.0], [7.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[8.0, -6.0], [8.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[9.0, -6.0], [9.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[10.0, -6.0], [10.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[11.0, -6.0], [11.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[12.0, -6.0], [12.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}]], "label": [[[0.0, 0.0], "D", "above left", {}], [[9.0, 5.5], "N", "above", {}], [[11.5, 3.0], "E", "right", {}], [[9.0, 0.5], "S", "below", {}], [[6.5, 3.0], "O", "left", {}], [[10.5, 4.5], "NE", "above right", {}], [[10.5, 1.5], "SE", "below right", {}], [[7.5, 1.5], "SO", "below left", {}], [[7.5, 4.5], "NO", "above left", {}]], "circle": [[[0.0, 0.0], 0.1, {}]], "path": [[[[9.0, 3.0], [9.5, 3.5], [9.0, 5.5], [9.0, 3.0]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [8.5, 3.5], [9.0, 5.5], [9.0, 3.0]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[9.5, 3.5], [10.5, 4.5], [10.0, 3.375], [9.5, 3.5]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.5, 3.5], [10.5, 4.5], [9.375, 4.0], [9.5, 3.5]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [8.5, 3.5], [6.5, 3.0], [9.0, 3.0]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [8.5, 2.5], [6.5, 3.0], [9.0, 3.0]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[8.5, 3.5], [7.5, 4.5], [8.625, 4.0], [8.5, 3.5]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[8.5, 3.5], [7.5, 4.5], [8.0, 3.375], [8.5, 3.5]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [8.5, 2.5], [9.0, 0.5], [9.0, 3.0]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [9.5, 2.5], [9.0, 0.5], [9.0, 3.0]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[8.5, 2.5], [7.5, 1.5], [8.0, 2.625], [8.5, 2.5]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[8.5, 2.5], [7.5, 1.5], [8.625, 2.0], [8.5, 2.5]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [9.5, 2.5], [11.5, 3.0], [9.0, 3.0]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [9.5, 3.5], [11.5, 3.0], [9.0, 3.0]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[9.5, 2.5], [10.5, 1.5], [9.375, 2.0], [9.5, 2.5]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.5, 2.5], [10.5, 1.5], [10.0, 2.625], [9.5, 2.5]], {"fill": "#808080", "opacity": "0.5", "stroke": null}]], "drawing_options": {"tools": ["segment"]}}

Exercice 2 : Donner les directions entre les différents points constituant une figure

Donner les directions pour tracer la figure du point A au point E.

On indiquera les directions par leur initiales et séparées de tirets.
S-NE-SE-NO signifiera par exemple "D'abord Sud ensuite Nord-Est puis Sud-Est et enfin Nord-Ouest".

{"init": {"range": [[-1.0, 16.0], [-1.0, 16.0]], "scale": [25, 25]}, "line": [[[0.0, 0.0], [15.0, 0.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 1.0], [15.0, 1.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 2.0], [15.0, 2.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 3.0], [15.0, 3.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 4.0], [15.0, 4.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 5.0], [15.0, 5.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 6.0], [15.0, 6.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 7.0], [15.0, 7.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 8.0], [15.0, 8.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 9.0], [15.0, 9.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 10.0], [15.0, 10.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 11.0], [15.0, 11.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 12.0], [15.0, 12.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 13.0], [15.0, 13.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 14.0], [15.0, 14.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 15.0], [15.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[0.0, 0.0], [0.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[1.0, 0.0], [1.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[2.0, 0.0], [2.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[3.0, 0.0], [3.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[4.0, 0.0], [4.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[5.0, 0.0], [5.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[6.0, 0.0], [6.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[7.0, 0.0], [7.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[8.0, 0.0], [8.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[9.0, 0.0], [9.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[10.0, 0.0], [10.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[11.0, 0.0], [11.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[12.0, 0.0], [12.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[13.0, 0.0], [13.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[14.0, 0.0], [14.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[15.0, 0.0], [15.0, 15.0], {"subtype": "segment", "stroke": "lightgrey"}], [[4.0, 8.0], [14.0, 5.0], {"subtype": "segment"}], [[14.0, 5.0], [14.0, 14.0], {"subtype": "segment"}], [[14.0, 14.0], [13.0, 4.0], {"subtype": "segment"}], [[13.0, 4.0], [14.0, 12.0], {"subtype": "segment"}]], "label": [[[4.0, 8.0], "A", "above", {}], [[14.0, 5.0], "B", "above", {}], [[14.0, 14.0], "C", "above", {}], [[13.0, 4.0], "D", "above", {}], [[14.0, 12.0], "E", "above", {}]], "circle": [[[4.0, 8.0], 0.15, {"fill": "black"}], [[14.0, 5.0], 0.15, {"fill": "black"}], [[14.0, 14.0], 0.15, {"fill": "black"}], [[13.0, 4.0], 0.15, {"fill": "black"}], [[14.0, 12.0], 0.15, {"fill": "black"}]]}

Exercice 3 : Donner les directions pour effectuer un parcours entre des villes sur une carte (Source OpenStreetMap)

En utilisant la carte de France ci-dessous, indiquer les directions pour effectuer le parcours de villes suivant : Partir de Reims puis aller à Nantes, ensuite à Saint-Martin-d'Hères et enfin Aulnay-sous-Bois.

On indiquera les directions par leur initiales et séparées de tirets.
NE-SE-NO signifiera par exemple "Nord-Est puis Sud-Est puis Nord-Ouest".

Exercice 4 : Tracer un itinéraire à partir d'une séquence de directions

À partir du point \(D\) tracer l'itinéraire correspondant aux instructions suivantes : \[O2 - N1 - E4 - N3 - E3 - S6\] Les chiffres indiquent une distance en nombre de carreaux.
Les lettres indiquent une direction (voir la rose des vents).

{"init": {"range": [[-7.0, 13.0], [-7.0, 7.0]], "scale": [35, 35]}, "line": [[[-6.0, -6.0], [12.0, -6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -5.0], [12.0, -5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -4.0], [12.0, -4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -3.0], [12.0, -3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -2.0], [12.0, -2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -1.0], [12.0, -1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 0.0], [12.0, 0.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 1.0], [12.0, 1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 2.0], [12.0, 2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 3.0], [12.0, 3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 4.0], [12.0, 4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 5.0], [12.0, 5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, 6.0], [12.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -6.0], [-6.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-5.0, -6.0], [-5.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-4.0, -6.0], [-4.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-3.0, -6.0], [-3.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.0, -6.0], [-2.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.0, -6.0], [-1.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[0.0, -6.0], [0.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[1.0, -6.0], [1.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[2.0, -6.0], [2.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.0, -6.0], [3.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[4.0, -6.0], [4.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[5.0, -6.0], [5.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[6.0, -6.0], [6.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[7.0, -6.0], [7.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[8.0, -6.0], [8.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[9.0, -6.0], [9.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[10.0, -6.0], [10.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[11.0, -6.0], [11.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[12.0, -6.0], [12.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}]], "label": [[[0.0, 0.0], "D", "above left", {}], [[6.5, 3.0], "O", "left", {}], [[9.0, 5.5], "N", "above", {}], [[9.0, 0.5], "S", "below", {}], [[11.5, 3.0], "E", "right", {}]], "circle": [[[0.0, 0.0], 0.1, {}]], "path": [[[[9.0, 3.0], [9.5, 3.5], [9.0, 5.5], [9.0, 3.0]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [8.5, 3.5], [9.0, 5.5], [9.0, 3.0]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [8.5, 3.5], [6.5, 3.0], [9.0, 3.0]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [8.5, 2.5], [6.5, 3.0], [9.0, 3.0]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [8.5, 2.5], [9.0, 0.5], [9.0, 3.0]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [9.5, 2.5], [9.0, 0.5], [9.0, 3.0]], {"fill": "#808080", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [9.5, 2.5], [11.5, 3.0], [9.0, 3.0]], {"fill": "#0000ff", "opacity": "0.5", "stroke": null}], [[[9.0, 3.0], [9.5, 3.5], [11.5, 3.0], [9.0, 3.0]], {"fill": "#808080", "opacity": "0.5", "stroke": null}]], "drawing_options": {"tools": ["segment"]}}

Exercice 5 : Tracer un itinéraire à partir d'une séquence de directions (avec diagonales)

À partir du point \(D\) tracer l'itinéraire correspondant aux instructions suivantes : \[E1 - NO3 - O3 - SE9 - NE3 - NO6 - N2\] Les chiffres indiquent une distance en nombre de carreaux. On considère qu'une diagonale compte pour \(\textit{1,5}\) coté de carreau.
Les lettres indiquent une direction (voir la rose des vents).

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