Géométrie plane et repérée - 2de

Projection orthogonale

Exercice 1 : Identifier une projection orthogonale et exprimer le cosinus et le sinus

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

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", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[0.0, 0.0], "G", "below left", {}], [[2.923076923076923, 4.384615384615385], "I", "above right", {}], [[2.923076923076923, 4.384615384615385], "I", "above right", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[0.0, 0.0], [2.923076923076923, 4.384615384615385], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[2.923076923076923, 4.384615384615385], [2.5902568053417854, 4.606495463105476], [2.368376726851694, 4.273675345370339], [2.701196844586831, 4.051795266880248], [2.923076923076923, 4.384615384615385]], {"stroke": "red"}]]}
2.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[1.0, 2.5], "I", "left", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[0.44, 1.3900000000000001], [0.56, 1.1099999999999999]], [[0.0, 0.0], [1.0, 2.5], {"subtype": "segment"}], [[1.44, 3.89], [1.56, 3.61]], [[1.0, 2.5], [2.0, 5.0], {"subtype": "segment"}], [[8.0, 1.0], [1.0, 2.5], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]]}
3.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[2.0, 5.0], "E", "above left", {}], [[2.5846153846153848, 0.3230769230769231], "I", "below", {}], [[2.5846153846153848, 0.3230769230769231], "I", "below", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [2.5846153846153848, 0.3230769230769231], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[2.5846153846153848, 0.3230769230769231], [2.1877042339299178, 0.2734630292412397], [2.138090340094234, 0.6703741799267069], [2.535001490779701, 0.7199880737623903], [2.5846153846153848, 0.3230769230769231]], {"stroke": "red"}]]}

Donner une expression simplifiée de \(\dfrac{sin(\widehat{GFE})}{IG}\) en fonction des longueurs de la figure.

Exercice 2 : Identifier une projection orthogonale et exprimer le cosinus et le sinus

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

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", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[0.0, 0.0], "G", "below left", {}], [[2.923076923076923, 4.384615384615385], "I", "above right", {}], [[2.923076923076923, 4.384615384615385], "I", "above right", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[0.0, 0.0], [2.923076923076923, 4.384615384615385], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[2.923076923076923, 4.384615384615385], [2.5902568053417854, 4.606495463105476], [2.368376726851694, 4.273675345370339], [2.701196844586831, 4.051795266880248], [2.923076923076923, 4.384615384615385]], {"stroke": "red"}]]}
2.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[1.0, 2.5], "I", "left", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[0.44, 1.3900000000000001], [0.56, 1.1099999999999999]], [[0.0, 0.0], [1.0, 2.5], {"subtype": "segment"}], [[1.44, 3.89], [1.56, 3.61]], [[1.0, 2.5], [2.0, 5.0], {"subtype": "segment"}], [[8.0, 1.0], [1.0, 2.5], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]]}
3.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[8.0, 1.0], "F", "right", {}], [[1.4482758620689655, 3.6206896551724137], "I", "left", {}], [[1.4482758620689655, 3.6206896551724137], "I", "left", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[8.0, 1.0], [1.4482758620689655, 3.6206896551724137], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[1.4482758620689655, 3.6206896551724137], [1.299719591527324, 3.24929897881831], [1.6711102678814278, 3.1007427082766683], [1.8196665384230692, 3.472133384630772], [1.4482758620689655, 3.6206896551724137]], {"stroke": "red"}]]}

Donner une expression simplifiée de \(EG \times sin(\widehat{GEF})\) en fonction des longueurs de la figure.

Exercice 3 : Identifier une projection orthogonale et exprimer le cosinus et le sinus

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", {}], [[1.0, 1.25], "I", "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"}], [[0.49, 0.715], [0.51, 0.535]], [[0.0, 0.0], [1.0, 1.25], {"subtype": "segment"}], [[1.49, 1.965], [1.51, 1.785]], [[1.0, 1.25], [2.0, 2.5], {"subtype": "segment"}], [[8.0, 1.0], [1.0, 1.25], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]]}
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", {}], [[0.0, 0.0], "G", "left", {}], [[0.7058823529411765, 2.823529411764706], "I", "above", {}], [[0.7058823529411765, 2.823529411764706], "I", "above", {}]], "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], [0.7058823529411765, 2.823529411764706], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}], [[0.0, 0.0], [0.7058823529411765, 2.823529411764706], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[0.7058823529411765, 2.823529411764706], [1.0939393529993091, 2.726515161750173], [0.9969251029847761, 2.33845816169204], [0.6088681029266433, 2.435472411706573], [0.7058823529411765, 2.823529411764706]], {"stroke": "red"}]]}
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", {}], [[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"}]]}

Donner une expression simplifiée de \(GF \times sin(\widehat{FGE})\) en fonction des longueurs de la figure.

Exercice 4 : Identifier une projection orthogonale et exprimer le cosinus et le sinus

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", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[0.0, 0.0], "G", "below left", {}], [[2.923076923076923, 4.384615384615385], "I", "above right", {}], [[2.923076923076923, 4.384615384615385], "I", "above right", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[0.0, 0.0], [2.923076923076923, 4.384615384615385], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[2.923076923076923, 4.384615384615385], [2.5902568053417854, 4.606495463105476], [2.368376726851694, 4.273675345370339], [2.701196844586831, 4.051795266880248], [2.923076923076923, 4.384615384615385]], {"stroke": "red"}]]}
2.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[2.0, 5.0], "E", "above left", {}], [[2.5846153846153848, 0.3230769230769231], "I", "below", {}], [[2.5846153846153848, 0.3230769230769231], "I", "below", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [2.5846153846153848, 0.3230769230769231], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[2.5846153846153848, 0.3230769230769231], [2.1877042339299178, 0.2734630292412397], [2.138090340094234, 0.6703741799267069], [2.535001490779701, 0.7199880737623903], [2.5846153846153848, 0.3230769230769231]], {"stroke": "red"}]]}
3.

{"init": {"range": [[0, 9], [-1, 7]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "G", "below left", {}], [[2.0, 5.0], "E", "above left", {}], [[8.0, 1.0], "F", "right", {}], [[8.0, 1.0], "F", "right", {}], [[1.4482758620689655, 3.6206896551724137], "I", "left", {}], [[1.4482758620689655, 3.6206896551724137], "I", "left", {}]], "line": [[[0.0, 0.0], [2.0, 5.0], {"subtype": "segment"}], [[0.0, 0.0], [8.0, 1.0], {"subtype": "segment"}], [[2.0, 5.0], [8.0, 1.0], {"subtype": "segment"}], [[8.0, 1.0], [1.4482758620689655, 3.6206896551724137], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[1.4482758620689655, 3.6206896551724137], [1.299719591527324, 3.24929897881831], [1.6711102678814278, 3.1007427082766683], [1.8196665384230692, 3.472133384630772], [1.4482758620689655, 3.6206896551724137]], {"stroke": "red"}]]}

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

Exercice 5 : Identifier une projection orthogonale et exprimer le cosinus et le sinus

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

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", {}], [[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": "- "}]]}
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", {}], [[0.0, 0.0], "G", "left", {}], [[0.7058823529411765, 2.823529411764706], "I", "above", {}], [[0.7058823529411765, 2.823529411764706], "I", "above", {}]], "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], [0.7058823529411765, 2.823529411764706], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}], [[0.0, 0.0], [0.7058823529411765, 2.823529411764706], {"subtype": "segment", "stroke": "blue", "stroke-dasharray": "- "}]], "path": [[[[0.7058823529411765, 2.823529411764706], [1.0939393529993091, 2.726515161750173], [0.9969251029847761, 2.33845816169204], [0.6088681029266433, 2.435472411706573], [0.7058823529411765, 2.823529411764706]], {"stroke": "red"}]]}
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 \(\dfrac{sin(\widehat{GFE})}{IG}\) en fonction des longueurs de la figure.

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