Exercices et devoirs de maths et physique-chimie sur mesure pour le collège et le lycée

On considère le triangle \(EFG\).
Le point \(I\) est le projeté orthogonal du point \(F\) sur la droite \((GE)\).

Dans quel(s) cas le point \(I\) est-il bien placé ?

1.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "left", {}], [[2.0, 2.5], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[8.0, 1.0], "F", "right", {}], [[3.6097560975609757, 4.512195121951219], "I", "above left", {}], [[3.6097560975609757, 4.512195121951219], "I", "above left", {}]], "line": [[[0.0, 0.0], [2.0, 2.5], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 2.5], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 2.5], [3.6097560975609757, 4.512195121951219], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}], [[8.0, 1.0], [3.6097560975609757, 4.512195121951219], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[3.6097560975609757, 4.512195121951219], [3.359878078539206, 4.199847598174007], [3.672225602316418, 3.949969579152237], [3.922103621338188, 4.262317102929449], [3.6097560975609757, 4.512195121951219]], {"stroke": "red"}]]}
2.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "left", {}], [[2.0, 2.5], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[4.0, 0.5], "I", "below right", {}]], "line": [[[0.0, 0.0], [2.0, 2.5], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 2.5], [8.0, 1.0], {"subtype": "segment"}], [[2.14, 0.42999999999999994], [1.86, 0.07000000000000003]], [[0.0, 0.0], [4.0, 0.5], {"subtype": "segment"}], [[6.14, 0.9299999999999999], [5.86, 0.5700000000000001]], [[4.0, 0.5], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 2.5], [4.0, 0.5], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]]}
3.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "left", {}], [[2.0, 2.5], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[5.0, 1.75], "I", "right", {}]], "line": [[[0.0, 0.0], [2.0, 2.5], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 2.5], [8.0, 1.0], {"subtype": "segment"}], [[3.65, 2.215], [3.35, 2.035]], [[2.0, 2.5], [5.0, 1.75], {"subtype": "segment"}], [[6.65, 1.465], [6.35, 1.285]], [[5.0, 1.75], [8.0, 1.0], {"subtype": "segment"}], [[0.0, 0.0], [5.0, 1.75], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]]}

Donner une expression simplifiée de \(\dfrac{cos(\widehat{FGE})}{GI}\) en fonction des longueurs de la figure.

Pour accéder à cet exercice, il faut être connecté.