Le plan - 5e

Hauteurs, médiatrices

Exercice 1 : Construire la médiatrice du côté d'un triangle

Dans le triangle \(ABC\) suivant, tracer la médiatrice du segment \([AB]\).

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Suivant Valider

Exercice 2 : Reconnaître une hauteur de triangle

Parmi les figures suivantes, pour lesquelles la droite \((DC)\) est-elle la hauteur du triangle \(ABC\) ?

1.

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2.

{"init": {"range": [[-3.972886424192989, 7.076965417556364], [-9.494818791996037, 1.0]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "C", "above", {}], [[-2.972886424192989, -8.494818791996037], "A", "below left", {}], [[6.076965417556364, -6.638560936966987], "B", "right", {}], [[1.552039496681687, -7.56668986448151], "D", "below", {}]], "path": [[[[0.0, 0.0], [-2.972886424192989, -8.494818791996037], [6.076965417556364, -6.638560936966987], [0.0, 0.0]], {"names": "CAB"}], [[[1.552039496681687, -7.56668986448151], [2.0418421468323382, -7.4662241294998815], [1.94137641185071, -6.976421479349231], [1.4515737617000586, -7.076887214330858], [1.552039496681687, -7.56668986448151]]]], "line": [[[-2.972886424192989, -8.494818791996037], [1.552039496681687, -7.56668986448151], {"subtype": "segment"}], [[1.552039496681687, -7.56668986448151], [6.076965417556364, -6.638560936966987], {"subtype": "segment"}], [[6.076965417556364, -6.638560936966987], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-2.972886424192989, -8.494818791996037], {"subtype": "segment"}], [[1.552039496681687, -7.56668986448151], [0.0, 0.0], {"subtype": "segment"}]], "arc": [[[6.076965417556364, -6.638560936966987], [0.5, 0.5], 132.47111094005993, 191.591434138463, false], [[-2.972886424192989, -8.494818791996037], [0.5, 0.5], 11.591434138463, 70.71175733686596, false]]}
3.

{"init": {"range": [[-1.0, 5.6011482061869025], [-4.300577197337564, 2.956894270190152]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "C", "left", {}], [[3.755820837635271, -3.300577197337564], "A", "below", {}], [[4.6011482061869025, 1.956894270190152], "B", "above right", {}], [[4.389816364048995, 0.6425264033082227], "D", "above right", {}]], "path": [[[[0.0, 0.0], [3.755820837635271, -3.300577197337564], [4.6011482061869025, 1.956894270190152], [0.0, 0.0]], {"names": "CAB"}]], "line": [[[3.755820837635271, -3.300577197337564], [4.389816364048995, 0.6425264033082227], {"subtype": "segment"}], [[4.389816364048995, 0.6425264033082227], [4.6011482061869025, 1.956894270190152], {"subtype": "segment"}], [[4.6011482061869025, 1.956894270190152], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [3.755820837635271, -3.300577197337564], {"subtype": "segment"}], [[4.389816364048995, 0.6425264033082227], [0.0, 0.0], {"subtype": "segment"}], [[4.122790358906563, 0.3792354756679521], [3.944773022144943, 0.20370819057443845]]], "arc": [[[4.389816364048995, 0.6425264033082227], [0.5, 0.5], 188.3271102794189, 260.8658227561957, false], [[4.6011482061869025, 1.956894270190152], [0.5, 0.5], 203.0403229676, 260.8658227561957, false], [[3.755820837635271, -3.300577197337564], [0.5, 0.5], 80.86582275619571, 138.69132254479132, false]]}
4.

{"init": {"range": [[-2.8260170334679353, 3.6941039045812314], [-1.0, 5.654638739309955]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "C", "below", {}], [[2.6941039045812314, 4.212102105994128], "A", "right", {}], [[-1.8260170334679353, 4.654638739309955], "B", "above left", {}], [[0.4340434355566481, 4.433370422652041], "D", "above", {}]], "path": [[[[0.0, 0.0], [2.6941039045812314, 4.212102105994128], [-1.8260170334679353, 4.654638739309955], [0.0, 0.0]], {"names": "CAB"}], [[[0.4340434355566481, 4.433370422652041], [-0.06357737015884402, 4.4820893483284605], [-0.11229629583526307, 3.9844685426129685], [0.385324509880229, 3.9357496169365493], [0.4340434355566481, 4.433370422652041]]]], "line": [[[2.6941039045812314, 4.212102105994128], [0.4340434355566481, 4.433370422652041], {"subtype": "segment"}], [[0.4340434355566481, 4.433370422652041], [-1.8260170334679353, 4.654638739309955], {"subtype": "segment"}], [[-1.8260170334679353, 4.654638739309955], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [2.6941039045812314, 4.212102105994128], {"subtype": "segment"}], [[0.4340434355566481, 4.433370422652041], [0.0, 0.0], {"subtype": "segment"}]], "arc": [[[-1.8260170334679353, 4.654638739309955], [0.5, 0.5], 291.42010123253954, 354.40835045554024, false], [[2.6941039045812314, 4.212102105994128], [0.5, 0.5], 174.40835045554024, 237.39659967854095, false]]}

Suivant Valider

Exercice 3 : Construire la hauteur (intérieure) du côté d'un triangle

Dans le triangle \( ABC \) suivant, tracer la hauteur relative au côté \( [AB] \).

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Suivant Valider

Exercice 4 : Reconnaître une médiatrice de segment

Parmi les figures suivantes, pour lesquelles la droite \((DC)\) est-elle la médiatrice du segment \([AB]\) ?

1.

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2.

{"path": [[[[0.0, 0.0], [-7.822715633357472, -9.099731870754468], [11.488319565650785, -3.466772787115607], [0.0, 0.0]], {"names": "CAB"}], [[[1.8328019661466568, -6.2832523289350375], [0.7855932446574838, -6.588719323292814], [0.4801262502997077, -5.54151060180364], [1.5273349717888807, -5.236043607445865], [1.8328019661466568, -6.2832523289350375]]]], "init": {"range": [[-8.822715633357472, 12.488319565650785], [-10.099731870754468, 1.0]], "scale": [30, 30]}, "arc": [[[-7.822715633357472, -9.099731870754468], [1.676318528500785, 1.676318528500785], 16.26177373265847, 49.315523834452904, false], [[-7.822715633357472, -9.099731870754468], [1.3410548228006283, 1.3410548228006283], 16.26177373265847, 49.315523834452904, false], [[-7.822715633357472, -9.099731870754468], [2.011582234200942, 2.011582234200942], 16.26177373265847, 49.315523834452904, false], [[11.488319565650785, -3.466772787115607], [1.676318528500785, 1.676318528500785], 163.2080236308641, 196.2617737326584, false], [[11.488319565650785, -3.466772787115607], [1.3410548228006283, 1.3410548228006283], 163.2080236308641, 196.2617737326584, false]], "line": [[[-7.822715633357472, -9.099731870754468], [1.8328019661466568, -6.2832523289350375], {"subtype": "segment"}], [[1.8328019661466568, -6.2832523289350375], [11.488319565650785, -3.466772787115607], {"subtype": "segment"}], [[11.488319565650785, -3.466772787115607], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-7.822715633357472, -9.099731870754468], {"subtype": "segment"}], [[1.8328019661466568, -6.2832523289350375], [0.0, 0.0], {"subtype": "segment"}]], "label": [[[0.0, 0.0], "C", "above", {}], [[-7.822715633357472, -9.099731870754468], "A", "below left", {}], [[11.488319565650785, -3.466772787115607], "B", "right", {}], [[1.8328019661466568, -6.2832523289350375], "D", "below", {}], [[-6.413478384753107, -8.19193553558835], 33, "above right"], [[9.812018980531244, -3.4590166719206605], 33, "left"]]}
3.

{"path": [[[[0.0, 0.0], [-6.378885849757456, -10.16414360956023], [3.2103753648498627, -7.327584186944911], [0.0, 0.0]], {"names": "CAB"}]], "init": {"range": [[-7.378885849757456, 4.210375364849863], [-11.16414360956023, 1.0]], "scale": [30, 30]}, "line": [[[-6.378885849757456, -10.16414360956023], [-1.5842552424537968, -8.745863898252571], {"subtype": "segment"}], [[-1.5842552424537968, -8.745863898252571], [3.2103753648498627, -7.327584186944911], {"subtype": "segment"}], [[3.2103753648498627, -7.327584186944911], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-6.378885849757456, -10.16414360956023], {"subtype": "segment"}], [[-1.5842552424537968, -8.745863898252571], [0.0, 0.0], {"subtype": "segment"}]], "label": [[[0.0, 0.0], "C", "above", {}], [[-6.378885849757456, -10.16414360956023], "A", "below left", {}], [[3.2103753648498627, -7.327584186944911], "B", "right", {}], [[-1.5842552424537968, -8.745863898252571], "D", "below", {}], [[-3.8113769807487072, -10.03035942678284], "5"], [[0.9832536265549522, -8.61207971547518], "5"], [[2.1547564964458, -3.423013941108716], "8"], [[-3.6976501053567397, -4.763127512292242], "12"]]}
4.

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Suivant Valider

Exercice 5 : Démontrer l'équidistance entre l'intersection de deux médiatrices et les sommets respectifs

Sachant que \((m)\) est la médiatrice de \([VX]\), \((m')\) est la médiatrice de \([XY]\) et que \(M\) est à l'intersection de \((m)\) et de \((m')\)

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Démontrer que \(MV = MY\).

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"Mediatrice": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "line", "check": "Line"}, {"type": "dummy_input", "fields": [{"text": "est la m\u00e9diatrice de", "type": "text"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "segment", "check": "Segment"}]}, "theorem_MediatriceEqualDistance": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Si un point appartient \u00e0 la", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "m\u00e9diatrice d'un segment alors", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "il est \u00e0 \u00e9gale distance des", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "extr\u00e9mit\u00e9s de ce segment.", "type": "text"}], "align": "left"}]}, "VarValuePointA": {"compound": {"blocks": {"VarValuePoint": {"output": "Point", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValuePoint", "value": "V"}}, "output": "Point", "colour": 330}, "SiAlorsImplication": {"previous_statement": true, "next_statement": true, "colour": 120, "tooltip": "Une implication", "inputs": [{"type": "dummy_input", "fields": [{"text": "Si", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "hypo", "check": null}, {"type": "dummy_input", "fields": [{"text": "alors,", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "concl", "check": null}]}, "VarValuePointM": {"compound": {"blocks": {"VarValuePoint": {"output": "Point", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValuePoint", "value": "M"}}, "output": "Point", "colour": 330}, "VarValueSegmentAB": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "[VX]"}}, "output": "Segment", "colour": 330}, "Start": {"next_statement": true, "colour": 65, "tooltip": "D\u00e9marrage de la d\u00e9monstration", "inputs": [{"type": "dummy_input", "fields": [{"text": "D\u00e9monstration", "type": "text"}], "align": "left"}], "creatable": false, "on_event": "start"}, "SimpleComparison": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": null}, {"type": "dummy_input", "fields": [{"name": "op", "values": [["=", "EQ"], ["\u2260", "NE"], ["<", "LT"], ["\u2264", "LE"], [">", "GT"], ["\u2265", "GE"], ["appartient \u00e0", "BELONGS"]], "type": "dropdown"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": null}]}, "Implication": {"previous_statement": true, "next_statement": true, "colour": 120, "tooltip": "\u00c0 partir d'hypoth\u00e8ses et d'une propri\u00e9t\u00e9 d\u00e9montre un r\u00e9sultat", "inputs": [{"type": "dummy_input", "fields": [{"text": "On sait que", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "hypo", "check": null}, {"type": "dummy_input", "fields": [{"text": "or,", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "theo", "check": null}, {"type": "dummy_input", "fields": [{"text": "donc", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "concl", "check": null}]}, "VarValueSegmentMB": {"compound": {"blocks": {"VarValueSegment": {"output": 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"segment_2", "check": "Segment"}]}, "VarValueLinem_prime": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(m')"}}, "output": "Line", "colour": 330}, "VarValueLinem": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(m)"}}, "output": "Line", "colour": 330}, "VarValueSegmentMC": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "MY"}}, "output": "Segment", "colour": 330}, "VarValueSegmentAC": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "VY"}}, "output": "Segment", "colour": 330}, "VarValuePointB": {"compound": {"blocks": {"VarValuePoint": {"output": "Point", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValuePoint", "value": "X"}}, "output": "Point", "colour": 330}}, "toolbox": [{"type": "And"}, {"type": "EqualityBetweenSegmentLength"}, {"type": "Implication"}, {"type": "Mediatrice"}, {"type": "SiAlorsImplication"}, {"type": "SimpleComparison", "op": "BELONGS"}, {"type": "VarValueLinem"}, {"type": "VarValueLinem_prime"}, {"type": "VarValuePointA"}, {"type": "VarValuePointB"}, {"type": "VarValuePointC"}, {"type": "VarValuePointM"}, {"type": "VarValueSegmentAB"}, {"type": "VarValueSegmentAC"}, {"type": "VarValueSegmentBC"}, {"type": "VarValueSegmentMA"}, {"type": "VarValueSegmentMB"}, {"type": "VarValueSegmentMC"}, {"type": "theorem_MediatriceEqualDistance"}]}, "ast": [[{"type": "Start", "deletable": false, "movable": false, "editable": false}]]}

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