Hauteurs, médiatrices

Le plan - Mathématiques 5e

Exercice 1 : Construire la médiatrice du côté d'un triangle à l'aide du quadrillage

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

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Exercice 2 : 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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Exercice 3 : Construire la hauteur (extérieure ou intérieure) du côté d'un triangle à l'aide du quadrillage

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

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Exercice 4 : Démontrer l'équidistance entre l'intersection de deux médiatrices et les sommets respectifs

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

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Démontrer que \(MQ = MS\).
Si plusieurs blocs "On sait que, or, donc" sont nécessaires, il faut les écrire à la suite les uns des autres et non imbriqués les uns dans les autres.

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"inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValuePoint", "value": "M"}}, "output": "Point", "colour": 330}, "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"}]}, "And": {"previous_statement": true, "next_statement": true, "colour": 120, "tooltip": "Combine plusieurs propositions", "inputs": [{"type": "statement_input", "fields": [], "align": "left", "name": "bool1", "check": null}, {"type": "dummy_input", "fields": [{"text": "et", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "bool2", "check": null}]}, "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}]}, "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": "QS"}}, "output": "Segment", "colour": 330}, "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": "Q"}}, "output": "Point", "colour": 330}, "VarValueSegmentBC": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "[RS]"}}, "output": "Segment", "colour": 330}, "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}, "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}]}, "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"}, "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": "MS"}}, "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": "R"}}, "output": "Point", "colour": 330}, "VarValuePointC": {"compound": {"blocks": {"VarValuePoint": {"output": "Point", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValuePoint", "value": "S"}}, "output": "Point", "colour": 330}, "VarValueSegmentMB": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "MR"}}, "output": "Segment", "colour": 330}}, "toolbox": [{"type": "And"}, {"type": "Implication"}, {"type": "Mediatrice"}, {"type": "SiAlorsImplication"}, {"type": "SimpleComparison", "op": "BELONGS"}, {"type": "SimpleComparison", "op": "EQ"}, {"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}]]}

Exercice 5 : Reconnaître une médiatrice d'un triangle isocèle

Sachant que MY = MZ, NY = NZ et \( (d) \) est la médiatrice de [YZ].

Prouver que la droite (MN) est la médiatrice du segment [YZ].

{"options": {"runner": {"environment": "theorems"}, "blocks": {"VarValueSegmentNA": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "NY"}}, "output": "Segment", "colour": 330}, "And": {"previous_statement": true, "next_statement": true, "colour": 120, "tooltip": "Combine plusieurs propositions", "inputs": [{"type": "statement_input", "fields": [], "align": "left", "name": "bool1", "check": null}, {"type": "dummy_input", "fields": [{"text": "et", "type": "text"}], "align": "left"}, {"type": "statement_input", "fields": [], "align": "left", "name": "bool2", "check": null}]}, "VarValueSegmentMA": {"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}, "VarValuePointN": {"compound": {"blocks": {"VarValuePoint": {"output": "Point", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValuePoint", "value": "N"}}, "output": "Point", "colour": 330}, "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}]}, "VarValueSegmentCD": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "[YZ]"}}, "output": "Segment", "colour": 330}, "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"}]}, "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"}, "DroiteConfondues": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "dummy_input", "fields": [{"text": "La droite ", "type": "text"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "line_1", "check": "Line"}, {"type": "dummy_input", "fields": [{"text": "et la droite ", "type": "text"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "line_2", "check": "Line"}, {"type": "dummy_input", "fields": [{"text": "sont confondues ", "type": "text"}], "align": "left"}]}, "theorem_EqualDistanceMediatrice": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Si un point est \u00e9quidistant", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "des extr\u00e9mit\u00e9s d'un segment", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "alors ce point appartient \u00e0 la", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "m\u00e9diatrice de ce segment.", "type": "text"}], "align": "left"}]}, "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}]}, "theorem_DeuxDroitesSontConfondues": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Par deux points, il ne passe", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "qu'une droite.", "type": "text"}], "align": "left"}]}, "VarValueLineD": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(d)"}}, "output": "Line", "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}]}, "VarValueSegmentMB": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "MZ"}}, "output": "Segment", "colour": 330}, "VarValueLineMN": {"compound": {"blocks": {"VarValueLine": {"output": "Line", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueLine", "value": "(MN)"}}, "output": "Line", "colour": 330}, "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}, "VarValueSegmentNB": {"compound": {"blocks": {"VarValueSegment": {"output": "Segment", "colour": 330, "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "VarValueSegment", "value": "NZ"}}, "output": "Segment", "colour": 330}, "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"}]}}, "toolbox": [{"type": "And"}, {"type": "DroiteConfondues"}, {"type": "DroiteConfondues", "line_1": {"type": "VarValueLineMN"}, "line_2": {"type": "VarValueLineD"}}, {"type": "Implication"}, {"type": "Mediatrice", "line": {"type": "VarValueLineD"}, "segment": {"type": "VarValueSegmentCD"}}, {"type": "Mediatrice"}, {"type": "SiAlorsImplication"}, {"type": "SimpleComparison", "op": "BELONGS"}, {"type": "SimpleComparison", "op": "EQ"}, {"type": "SimpleComparison", "value_1": {"type": "VarValuePointM"}, "value_2": {"type": "VarValueLineD"}, "op": "BELONGS"}, {"type": "VarValueLineD"}, {"type": "VarValueLineMN"}, {"type": "VarValuePointM"}, {"type": "VarValuePointN"}, {"type": "VarValueSegmentCD"}, {"type": "VarValueSegmentMA"}, {"type": "VarValueSegmentMB"}, {"type": "VarValueSegmentNA"}, {"type": "VarValueSegmentNB"}, {"type": "theorem_DeuxDroitesSontConfondues"}, {"type": "theorem_EqualDistanceMediatrice"}, {"type": "theorem_MediatriceEqualDistance"}]}, "ast": [[{"type": "Start", "deletable": false, "movable": false, "editable": false}]]}

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