Proportionnalité

Pour aller plus loin (Ancien programme) - Mathématiques 5e

Exemple d'exercice parmi les 105 exercices du chapitre

On considère la figure modèle suivante :

{"path": [[[[0.0, 5.9], [0.0, 0.0], [5.719, 1.45], [5.719, 7.3500000000000005], [0.0, 5.9]]], [[[2.8595, 3.6750000000000003], [2.3829166666666666, 4.045833333333333], [2.0120871398423366, 3.569247038149066], [2.48867047317567, 3.198413704815733], [2.8595, 3.6750000000000003]]]], "init": {"range": [[-1.0, 6.719], [-1.0, 8.350000000000001]], "scale": [45, 45]}, "arc": [[[0.0, 0.0], [0.9833257203095128, 0.9833257203095128], 14.227024565873648, 90.0, false], [[5.719, 7.3500000000000005], [0.9833257203095128, 0.9833257203095128], 194.22702456587365, 270.0, false], [[5.719, 1.45], [0.9833257203095128, 0.9833257203095128], 90.0, 194.22702456587365, false], [[5.719, 1.45], [0.7866605762476103, 0.7866605762476103], 90.0, 194.22702456587365, false], [[0.0, 5.9], [0.9833257203095128, 0.9833257203095128], 270.0, 14.227024565873648, false], [[0.0, 5.9], [0.7866605762476103, 0.7866605762476103], 270.0, 14.227024565873648, false]], "line": [[[0.19666666666666668, 2.7533333333333334], [-0.19666666666666668, 3.146666666666667]], [[0.19666666666666668, 2.95], [-0.19666666666666668, 3.3433333333333337]], [[0.19666666666666668, 2.5566666666666666], [-0.19666666666666668, 2.95]], [[0.0, 5.9], [0.0, 0.0], {"subtype": "segment"}], [[3.0018011017034003, 0.963968516774813], [2.7171988982966, 0.48603148322518697]], [[2.8111662924642937, 0.9156348092391067], [2.5265640890574934, 0.43769777568948065]], [[3.192435910942507, 1.0123022243105193], [2.9078337075357066, 0.5343651907608933]], [[0.0, 0.0], [5.719, 1.45], {"subtype": "segment"}], [[5.915666666666667, 4.203333333333334], [5.522333333333334, 4.596666666666667]], [[5.915666666666667, 4.4], [5.522333333333334, 4.793333333333333]], [[5.915666666666667, 4.006666666666668], [5.522333333333334, 4.4]], [[5.719, 1.45], [5.719, 7.3500000000000005], {"subtype": "segment"}], [[2.7171988982966, 6.386031483225187], [3.0018011017034003, 6.863968516774813]], [[2.9078337075357066, 6.434365190760893], [3.192435910942507, 6.912302224310519]], [[2.5265640890574934, 6.337697775689481], [2.8111662924642937, 6.815634809239107]], [[5.719, 7.3500000000000005], [0.0, 5.9], {"subtype": "segment"}], [[0.0, 5.9], [5.719, 1.45], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}], [[0.0, 0.0], [5.719, 7.3500000000000005], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}]], "label": [[[0.6038594303094982, 0.776069108165317], "76\u00b0", "above right"], [[5.115140569690502, 6.573930891834683], "76\u00b0", "below left"], [[4.942930891834683, 2.053859430309498], "104\u00b0", "above left"], [[0.7760691081653174, 5.296140569690503], "104\u00b0", "below right"]]}

Parmi les figures A, B, C, D ci-dessous, lesquelles sont des agrandissements ou des réductions de la figure modèle.

A.

{"path": [[[[0.0, 4.130000000000001], [0.0, 0.0], [4.0033, 1.0150000000000001], [4.0033, 5.1450000000000005], [0.0, 4.130000000000001]]], [[[2.00165, 2.5725000000000002], [1.6680416666666669, 2.8320833333333337], [1.4084609978896356, 2.498472926704346], [1.7420693312229691, 2.238889593371013], [2.00165, 2.5725000000000002]]]], "init": {"range": [[-1.0, 5.0033], [-1.0, 6.1450000000000005]], "scale": [45, 45]}, "arc": [[[0.0, 0.0], [0.688328004216659, 0.688328004216659], 14.227024565873648, 90.0, false], [[4.0033, 5.1450000000000005], [0.688328004216659, 0.688328004216659], 194.22702456587365, 270.0, false], [[4.0033, 1.0150000000000001], [0.688328004216659, 0.688328004216659], 90.0, 194.22702456587365, false], [[4.0033, 1.0150000000000001], [0.5506624033733272, 0.5506624033733272], 90.0, 194.22702456587365, false], [[0.0, 4.130000000000001], [0.688328004216659, 0.688328004216659], 270.0, 14.227024565873648, false], [[0.0, 4.130000000000001], [0.5506624033733272, 0.5506624033733272], 270.0, 14.227024565873648, false]], "line": [[[0.13666666666666666, 1.9283333333333337], [-0.13666666666666666, 2.201666666666667]], [[0.13666666666666666, 2.0650000000000004], [-0.13666666666666666, 2.338333333333334]], [[0.13666666666666666, 1.791666666666667], [-0.13666666666666666, 2.0650000000000004]], [[0.0, 4.130000000000001], [0.0, 0.0], {"subtype": "segment"}], [[2.1005372062684646, 0.6735628675892769], [1.9027627937315355, 0.34143713241072327]], [[1.968062169339594, 0.6399750369288708], [1.7702877568026647, 0.3078493017503171]], [[2.233012243197335, 0.707150698249683], [2.035237830660406, 0.3750249630711294]], [[0.0, 0.0], [4.0033, 1.0150000000000001], {"subtype": "segment"}], [[4.139966666666667, 2.9433333333333334], [3.8666333333333336, 3.216666666666667]], [[4.139966666666667, 3.08], [3.8666333333333336, 3.3533333333333335]], [[4.139966666666667, 2.8066666666666666], [3.8666333333333336, 3.08]], [[4.0033, 1.0150000000000001], [4.0033, 5.1450000000000005], {"subtype": "segment"}], [[1.9027627937315355, 4.471437132410724], [2.1005372062684646, 4.803562867589278]], [[2.035237830660406, 4.50502496307113], [2.233012243197335, 4.837150698249684]], [[1.7702877568026647, 4.437849301750318], [1.968062169339594, 4.769975036928872]], [[4.0033, 5.1450000000000005], [0.0, 4.130000000000001], {"subtype": "segment"}], [[0.0, 4.130000000000001], [4.0033, 1.0150000000000001], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}], [[0.0, 0.0], [4.0033, 5.1450000000000005], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}]], "label": [[[0.42270160121664874, 0.5432483757157219], "76\u00b0", "above right"], [[3.5805983987833514, 4.601751624284279], "76\u00b0", "below left"], [[3.4600516242842785, 1.4377016012166486], "104\u00b0", "above left"], [[0.543248375715722, 3.7072983987833523], "104\u00b0", "below right"]]}
B.

{"path": [[[[0.0, 7.670000000000001], [0.0, 0.0], [7.4347, 1.885], [7.4347, 9.555000000000001], [0.0, 7.670000000000001]]], [[[3.71735, 4.777500000000001], [3.0977916666666667, 5.259583333333334], [2.6157132817950375, 4.640021149593786], [3.2352716151283714, 4.1579378162604534], [3.71735, 4.777500000000001]]]], "init": {"range": [[-1.0, 8.4347], [-1.0, 10.555000000000001]], "scale": [45, 45]}, "arc": [[[0.0, 0.0], [1.2783234364023668, 1.2783234364023668], 14.227024565873648, 90.0, false], [[7.4347, 9.555000000000001], [1.2783234364023668, 1.2783234364023668], 194.22702456587365, 270.0, false], [[7.4347, 1.885], [1.2783234364023668, 1.2783234364023668], 90.0, 194.22702456587365, false], [[7.4347, 1.885], [1.0226587491218935, 1.0226587491218935], 90.0, 194.22702456587365, false], [[0.0, 7.670000000000001], [1.2783234364023668, 1.2783234364023668], 270.0, 14.227024565873648, false], [[0.0, 7.670000000000001], [1.0226587491218935, 1.0226587491218935], 270.0, 14.227024565873648, false]], "line": [[[0.25666666666666665, 3.5783333333333336], [-0.25666666666666665, 4.091666666666667]], [[0.0, 7.670000000000001], [0.0, 0.0], {"subtype": "segment"}], [[3.903064997138336, 1.2543741659603491], [3.5316350028616643, 0.6306258340396509]], [[0.0, 0.0], [7.4347, 1.885], {"subtype": "segment"}], [[7.691366666666667, 5.463333333333334], [7.1780333333333335, 5.9766666666666675]], [[7.4347, 1.885], [7.4347, 9.555000000000001], {"subtype": "segment"}], [[3.5316350028616643, 8.300625834039652], [3.903064997138336, 8.92437416596035]], [[7.4347, 9.555000000000001], [0.0, 7.670000000000001], {"subtype": "segment"}], [[0.0, 7.670000000000001], [7.4347, 1.885], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}], [[0.0, 0.0], [7.4347, 9.555000000000001], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}]], "label": [[[0.7850172594023478, 1.0088898406149123], "76\u00b0", "above right"], [[6.649682740597654, 8.546110159385089], "76\u00b0", "below left"], [[6.4258101593850885, 2.6700172594023477], "104\u00b0", "above left"], [[1.0088898406149123, 6.884982740597653], "104\u00b0", "below right"]]}
C.

{"path": [[[[0.0, 3.540000000000001], [0.0, 0.0], [3.431400000000001, 0.8700000000000001], [3.431400000000001, 4.410000000000001], [0.0, 3.540000000000001]]], [[[1.7157000000000004, 2.2050000000000005], [1.4297500000000003, 2.4275000000000007], [1.2072522839054023, 2.14154822288944], [1.4932022839054024, 1.9190482228894399], [1.7157000000000004, 2.2050000000000005]]]], "init": {"range": [[-1.0, 4.431400000000001], [-1.0, 5.410000000000001]], "scale": [45, 45]}, "arc": [[[0.0, 0.0], [0.5899954321857078, 0.5899954321857078], 14.227024565873648, 90.0, false], [[3.431400000000001, 4.410000000000001], [0.5899954321857078, 0.5899954321857078], 194.22702456587365, 270.0, false], [[3.431400000000001, 0.8700000000000001], [0.5899954321857078, 0.5899954321857078], 90.0, 194.22702456587365, false], [[3.431400000000001, 0.8700000000000001], [0.4719963457485663, 0.4719963457485663], 90.0, 194.22702456587365, false], [[0.0, 3.540000000000001], [0.5899954321857078, 0.5899954321857078], 270.0, 14.227024565873648, false], [[0.0, 3.540000000000001], [0.4719963457485663, 0.4719963457485663], 270.0, 14.227024565873648, false]], "line": [[[0.11666666666666667, 1.6533333333333338], [-0.11666666666666667, 1.8866666666666672]], [[0.11666666666666667, 1.7700000000000005], [-0.11666666666666667, 2.003333333333334]], [[0.0, 3.540000000000001], [0.0, 0.0], {"subtype": "segment"}], [[1.8001159077901532, 0.5767609845274315], [1.6312840922098477, 0.2932390154725686]], [[1.6870274616313612, 0.5480884461587922], [1.5181956460510557, 0.2645664771039292]], [[0.0, 0.0], [3.431400000000001, 0.8700000000000001], {"subtype": "segment"}], [[3.5480666666666676, 2.523333333333334], [3.314733333333334, 2.7566666666666673]], [[3.5480666666666676, 2.6400000000000006], [3.314733333333334, 2.873333333333334]], [[3.431400000000001, 0.8700000000000001], [3.431400000000001, 4.410000000000001], {"subtype": "segment"}], [[1.6312840922098477, 3.8332390154725697], [1.8001159077901532, 4.116760984527432]], [[1.7443725383686397, 3.861911553841209], [1.9132043539489452, 4.145433522896072]], [[3.431400000000001, 4.410000000000001], [0.0, 3.540000000000001], {"subtype": "segment"}], [[0.0, 3.540000000000001], [3.431400000000001, 0.8700000000000001], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}], [[0.0, 0.0], [3.431400000000001, 4.410000000000001], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}]], "label": [[[0.362315658185699, 0.46564146489919034], "76\u00b0", "above right"], [[3.0690843418143015, 3.944358535100811], "76\u00b0", "below left"], [[2.9657585351008104, 1.232315658185699], "104\u00b0", "above left"], [[0.4656414648991902, 3.177684341814302], "104\u00b0", "below right"]]}
D.

{"path": [[[[0.0, 5.09], [0.0, 0.0], [3.902, 3.268], [3.902, 8.358], [0.0, 5.09]]], [[[1.951, 4.179], [1.6258333333333335, 4.3308333333333335], [1.4740218732969645, 4.005656454044772], [1.7991885399636314, 3.853823120711438], [1.951, 4.179]]]], "init": {"range": [[-1.0, 4.902], [-1.0, 9.358]], "scale": [45, 45]}, "arc": [[[0.0, 0.0], [0.8482895862983479, 0.8482895862983479], 39.9468212184741, 90.0, false], [[3.902, 8.358], [0.848289586298348, 0.848289586298348], 219.94682121847404, 270.0, false], [[3.902, 3.268], [0.8482895862983479, 0.8482895862983479], 90.0, 219.94682121847404, false], [[3.902, 3.268], [0.6786316690386784, 0.6786316690386784], 90.0, 219.94682121847404, false], [[0.0, 5.09], [0.848289586298348, 0.848289586298348], 270.0, 39.9468212184741, false], [[0.0, 5.09], [0.6786316690386784, 0.6786316690386784], 270.0, 39.9468212184741, false]], "line": [[[0.16999999999999998, 2.375], [-0.16999999999999998, 2.715]], [[0.16999999999999998, 2.545], [-0.16999999999999998, 2.885]], [[0.16999999999999998, 2.205], [-0.16999999999999998, 2.545]], [[0.0, 5.09], [0.0, 0.0], {"subtype": "segment"}], [[1.972175944657908, 1.8734818977873802], [1.9298240553420922, 1.3945181022126196]], [[1.841847023435264, 1.764328921222644], [1.7994951341194483, 1.2853651256478835]], [[2.102504865880552, 1.9826348743521163], [2.060152976564736, 1.5036710787773557]], [[0.0, 0.0], [3.902, 3.268], {"subtype": "segment"}], [[4.072, 5.643000000000001], [3.732, 5.9830000000000005]], [[4.072, 5.813000000000001], [3.732, 6.1530000000000005]], [[4.072, 5.473000000000001], [3.732, 5.813000000000001]], [[3.902, 3.268], [3.902, 8.358], {"subtype": "segment"}], [[1.9298240553420922, 6.48451810221262], [1.972175944657908, 6.96348189778738]], [[2.060152976564736, 6.593671078777357], [2.102504865880552, 7.072634874352117]], [[1.7994951341194483, 6.375365125647884], [1.841847023435264, 6.854328921222644]], [[3.902, 8.358], [0.0, 5.09], {"subtype": "segment"}], [[0.0, 5.09], [3.902, 3.268], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}], [[0.0, 0.0], [3.902, 8.358], {"opacity": 0.3, "subtype": "segment", "stroke-dasharray": "- "}]], "label": [[[0.35885941598699583, 0.768645003743402], "50\u00b0", "above right"], [[3.5431405840130044, 7.589354996256598], "50\u00b0", "below left"], [[3.1333549962565983, 3.626859415986996], "130\u00b0", "above left"], [[0.7686450037434019, 4.731140584013003], "130\u00b0", "below right"]]}

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