Le plan - 5e

Hauteurs, médiatrices

Exercice 1 : Démontrer qu'une médiatrice et une hauteur sont parallèles

Sachant que \((h)\) est la hauteur issue de \(Z\) et \((m)\) est la médiatrice de \([XY]\)

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Démontrer que \((h)\) et \((m)\) sont parallèles.
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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Exercice 2 : Démontrer l'équidistance entre un point sur la médiatrice et les deux sommets du triangle associés

Sachant que \((d)\) est la médiatrice de \([GH]\) et \(M \in (d)\)

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Prouver que \(GM = HM\).

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["\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}]}, "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"}]}, "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": "[GI]"}}, "output": "Segment", "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}, "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}, "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": "G"}}, "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"}]}, "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": "HI"}}, "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": "H"}}, "output": "Point", "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"}]}}, "toolbox": [{"type": "And"}, {"type": "Implication"}, {"type": "Mediatrice"}, {"type": "SimpleComparison", "op": "BELONGS"}, {"type": "SimpleComparison", "op": "EQ"}, {"type": "VarValueLined"}, {"type": "VarValuePointA"}, {"type": "VarValuePointB"}, {"type": "VarValuePointC"}, {"type": "VarValuePointM"}, {"type": "VarValueSegmentAB"}, {"type": "VarValueSegmentAC"}, {"type": "VarValueSegmentAM"}, {"type": "VarValueSegmentBC"}, {"type": "VarValueSegmentBM"}, {"type": "theorem_EqualDistanceMediatrice"}, {"type": "theorem_MediatriceEqualDistance"}]}, "ast": [[{"type": "Start", "deletable": false, "movable": false, "editable": false}]]}

Exercice 3 : Reconnaître une médiatrice dans un triangle

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

1.

{"init": {"range": [[-1.0, 11.391282989122638], [-2.976550398949289, 7.001769559052604]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "C", "left", {}], [[9.802716384778936, -1.9765503989492894], "A", "below right", {}], [[10.391282989122638, 6.001769559052604], "B", "above right", {}], [[9.894679916707641, -0.729937905511493], "D", "below right", {}], [[9.676785716315752, -1.8105858108041268], 83, "above left"], [[9.79338180243483, 5.048806196190573], 56, "below left"]], "path": [[[[0.0, 0.0], [9.802716384778936, -1.9765503989492894], [10.391282989122638, 6.001769559052604], [0.0, 0.0]], {"names": "CAB"}]], "line": [[[9.802716384778936, -1.9765503989492894], [9.894679916707641, -0.729937905511493], {"subtype": "segment"}], [[9.894679916707641, -0.729937905511493], [10.391282989122638, 6.001769559052604], {"subtype": "segment"}], [[10.391282989122638, 6.001769559052604], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [9.802716384778936, -1.9765503989492894], {"subtype": "segment"}], [[9.894679916707641, -0.729937905511493], [0.0, 0.0], {"subtype": "segment"}], [[9.708268383431548, -1.852076957840418], [9.645303049199955, -1.7690946637678369]]], "arc": [[[0.0, 0.0], [1.6535945694153693, 1.6535945694153693], 348.60013446075686, 355.78089024221526, false], [[0.0, 0.0], [1.3228756555322956, 1.3228756555322956], 348.60013446075686, 355.78089024221526, false], [[0.0, 0.0], [1.6535945694153693, 1.6535945694153693], 355.78089024221526, 30.009756570027776, false], [[0.0, 0.0], [1.3228756555322956, 1.3228756555322956], 355.78089024221526, 30.009756570027776, false], [[0.0, 0.0], [1.984313483298443, 1.984313483298443], 355.78089024221526, 30.009756570027776, false], [[9.802716384778936, -1.9765503989492894], [0.20833333333333348, 0.20833333333333348], 85.78089024221515, 168.60013446075698, false], [[10.391282989122638, 6.001769559052604], [1.1249999999999998, 1.1249999999999998], 210.00975657002778, 265.78089024221526, false]]}
2.

{"init": {"range": [[-2.9787600761352433, 5.996887510796912], [-1.0, 5.591787077064141]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "C", "below left", {}], [[4.996887510796912, 0.1763950238579211], "A", "right", {}], [[-1.9787600761352433, 4.591787077064141], "B", "above left", {}], [[1.5090637173308346, 2.384091050461031], "D", "above right", {}]], "path": [[[[0.0, 0.0], [4.996887510796912, 0.1763950238579211], [-1.9787600761352433, 4.591787077064141], [0.0, 0.0]], {"names": "CAB"}], [[[1.5090637173308346, 2.384091050461031], [1.9064122257410063, 2.132580430905892], [1.6549016061858675, 1.73523192249572], [1.2575530977756955, 1.9867425420508593], [1.5090637173308346, 2.384091050461031]]]], "line": [[[4.996887510796912, 0.1763950238579211], [1.5090637173308346, 2.384091050461031], {"subtype": "segment"}], [[1.5090637173308346, 2.384091050461031], [-1.9787600761352433, 4.591787077064141], {"subtype": "segment"}], [[-1.9787600761352433, 4.591787077064141], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [4.996887510796912, 0.1763950238579211], {"subtype": "segment"}], [[1.5090637173308346, 2.384091050461031], [0.0, 0.0], {"subtype": "segment"}]], "arc": [[[4.996887510796912, 0.1763950238579211], [0.6879687605986525, 0.6879687605986525], 147.66732059457763, 182.02175760949467, false], [[-1.9787600761352433, 4.591787077064141], [0.6879687605986525, 0.6879687605986525], 293.3128835796606, 327.66732059457763, false]]}
3.

{"init": {"range": [[-8.573316155418848, 1.0], [-1.0, 8.599295753813891]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "C", "below right", {}], [[-2.500140805248013, 7.59929575381389], "A", "above", {}], [[-7.573316155418848, 2.577766942543852], "B", "left", {}], [[-5.03672848033343, 5.088531348178871], "D", "above left", {}], [[-3.979990598400213, 0.7208847596155122], "8"], [[-0.6801232210879649, 3.987158437300546], "8"]], "path": [[[[0.0, 0.0], [-2.500140805248013, 7.59929575381389], [-7.573316155418848, 2.577766942543852], [0.0, 0.0]], {"names": "CAB"}]], "line": [[[-3.7698817041150354, 6.061074540226078], [-3.7669875814664078, 6.626752561766684]], [[-3.627738668067727, 6.2017705149490725], [-3.624844545419099, 6.767448536489678]], [[-3.912024740162344, 5.920378565503083], [-3.9091306175137164, 6.486056587043689]], [[-2.500140805248013, 7.59929575381389], [-5.03672848033343, 5.088531348178871], {"subtype": "segment"}], [[-6.306469379200453, 3.550310134591059], [-6.303575256551825, 4.115988156131665]], [[-6.164326343153145, 3.6910061093140536], [-6.161432220504516, 4.2566841308546595]], [[-6.448612415247761, 3.409614159868064], [-6.445718292599133, 3.97529218140867]], [[-5.03672848033343, 5.088531348178871], [-7.573316155418848, 2.577766942543852], {"subtype": "segment"}], [[-7.573316155418848, 2.577766942543852], [0.0, 0.0], {"subtype": "segment"}], [[0.0, 0.0], [-2.500140805248013, 7.59929575381389], {"subtype": "segment"}], [[-5.03672848033343, 5.088531348178871], [0.0, 0.0], {"subtype": "segment"}]]}
4.

{"init": {"range": [[-9.41931765374475, 1.0], [-11.444362193155895, 1.0]], "scale": [30, 30]}, "label": [[[0.0, 0.0], "C", "above right", {}], [[-8.41931765374475, -7.079201243455506], "A", "left", {}], [[-3.4518543390728076, -10.444362193155895], "B", "below", {}], [[-5.935585996408779, -8.7617817183057], "D", "below left", {}], [[-7.513967920046804, -8.417237812347798], "3"], [[-5.030236262710833, -10.099818287197992], "3"]], "path": [[[[0.0, 0.0], [-8.41931765374475, -7.079201243455506], [-3.4518543390728076, -10.444362193155895], [0.0, 0.0]], {"names": "CAB"}]], "line": [[[-8.41931765374475, -7.079201243455506], [-5.935585996408779, -8.7617817183057], {"subtype": "segment"}], [[-5.935585996408779, -8.7617817183057], [-3.4518543390728076, -10.444362193155895], {"subtype": "segment"}], [[-1.8530636759742782, -4.969522614173789], [-1.5987906630985294, -5.474839578982106]], [[-3.4518543390728076, -10.444362193155895], [0.0, 0.0], {"subtype": "segment"}], [[-4.234024579786725, -3.8213918744041213], [-4.185293073958025, -3.2578093690513845]], [[-4.080946076991366, -3.6926791245231123], [-4.032214571162666, -3.1290966191703755]], [[-4.387103082582084, -3.9501046242851303], [-4.338371576753384, -3.3865221189323935]], [[0.0, 0.0], [-8.41931765374475, -7.079201243455506], {"subtype": "segment"}], [[-5.935585996408779, -8.7617817183057], [0.0, 0.0], {"subtype": "segment"}]]}

Exercice 4 : Construire la hauteur (extérieure) du côté d'un triangle

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

{"init": {"range": [[-10, 10], [-10, 10]], "scale": [35, 35]}, "line": [[[-10.0, -10.0], [10.0, -10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -9.0], [10.0, -9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -8.0], [10.0, -8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -7.0], [10.0, -7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -6.0], [10.0, -6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -5.0], [10.0, -5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -4.0], [10.0, -4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -3.0], [10.0, -3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -2.0], [10.0, -2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -1.0], [10.0, -1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 0.0], [10.0, 0.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 1.0], [10.0, 1.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 2.0], [10.0, 2.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 3.0], [10.0, 3.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 4.0], [10.0, 4.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 5.0], [10.0, 5.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 6.0], [10.0, 6.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 7.0], [10.0, 7.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 8.0], [10.0, 8.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 9.0], [10.0, 9.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, 10.0], [10.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-10.0, -10.0], [-10.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-9.0, -10.0], [-9.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-8.0, -10.0], [-8.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-7.0, -10.0], [-7.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-6.0, -10.0], [-6.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-5.0, -10.0], [-5.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-4.0, -10.0], [-4.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-3.0, -10.0], [-3.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-2.0, -10.0], [-2.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-1.0, -10.0], [-1.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[0.0, -10.0], [0.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[1.0, -10.0], [1.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[2.0, -10.0], [2.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[3.0, -10.0], [3.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[4.0, -10.0], [4.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[5.0, -10.0], [5.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[6.0, -10.0], [6.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[7.0, -10.0], [7.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[8.0, -10.0], [8.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[9.0, -10.0], [9.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[10.0, -10.0], [10.0, 10.0], {"subtype": "segment", "stroke": "#ccc"}], [[-7.1, -5.0], [-6.9, -5.0], {}], [[-7.0, -5.1], [-7.0, -4.9], {}], [[-7.1, -1.0], [-6.9, -1.0], {}], [[-7.0, -1.1], [-7.0, -0.9], {}], [[-5.1, 3.0], [-4.9, 3.0], {}], [[-5.0, 2.9], [-5.0, 3.1], {}], [[-7.0, -5.0], [-7.0, -1.0], {"subtype": "segment"}], [[-7.0, -1.0], [-5.0, 3.0], {"subtype": "segment"}], [[-5.0, 3.0], [-7.0, -5.0], {"subtype": "segment"}]], "label": [[[-7.0, -5.0], "A", "below", {}], [[-7.0, -1.0], "B", "left", {}], [[-5.0, 3.0], "C", "above", {}]], "drawing_options": {"tools": ["line"]}}

Exercice 5 : 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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