{"id":15296,"date":"2023-02-02T11:10:16","date_gmt":"2023-02-02T10:10:16","guid":{"rendered":"https:\/\/college-willy-ronis.fr\/maths\/?p=15296"},"modified":"2025-12-22T08:32:10","modified_gmt":"2025-12-22T07:32:10","slug":"chapitre-2-le-theoreme-de-pythagore-2","status":"publish","type":"post","link":"https:\/\/college-willy-ronis.fr\/maths\/chapitre-2-le-theoreme-de-pythagore-2\/","title":{"rendered":"Chapitre 2 : Le th\u00e9or\u00e8me de Pythagore"},"content":{"rendered":"\t\t
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Chapitre 2<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Le th\u00e9or\u00e8me de Pythagore<\/h1>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Chapitre 2 : Le th\u00e9or\u00e8me de Pythagore<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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A) Racine carr\u00e9 d\u2019un nombre positif<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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\n\t\t\t\t\t\t\tD\u00e9finition :<\/strong> $a$ d\u00e9signe un nombre positif. La racine carr\u00e9e<\/strong> de $a$ est le nombre positif dont le carr\u00e9 est $a$. Ce nombre est not\u00e9e $\\sqrt{a}$ (lire \u00ab\u00a0racine carr\u00e9e de $a$\u00a0\u00bb.)\nAinsi,\n\\[\\sqrt{a}\\geq 0~~~et~~~\\left(\\sqrt{a}\\right)^{2}=a\\]\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Exemple :<\/strong> Quelques carr\u00e9s parfaits \u00e0 conna\u00eetre :<\/p>

\"\"<\/p>

Donc $\\sqrt{0}=0$ ; $\\sqrt{1}=1$ ; $\\sqrt{9}=3$ ; $\\sqrt{16}=4$ ; $\\sqrt{25}=5$…<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Remarque :<\/strong> Il n’existe pas de nombre entier positif dont le carr\u00e9 vaut $27$. Or :
\\[5^{2}\\leq 27\\leq 6^{2}\\]Donc $\\sqrt{27}$ est comprise entre $5$ et $6$. Pour obtenir une valeur approch\u00e9e de $\\sqrt{27}$, on utilise la calculatrice :
$\\sqrt{27}\\approx 5,2$.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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B) L'\u00e9galit\u00e9 de Pythagore<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Propri\u00e9t\u00e9 :<\/strong> Si un triangle est rectangle alors le carr\u00e9 de la longueur du plus grand c\u00f4t\u00e9 est \u00e9gal \u00e0 la somme des carr\u00e9s des longueurs des deux autres c\u00f4t\u00e9s.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Exemple :<\/strong><\/p>

\"\"
L’\u00e9galit\u00e9 de Pythagore : $\\text{AC}^{2}=\\text{AB}^{2}+\\text{BC}^{2}$<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Vocabulaire :<\/strong> Dans un triangle rectangle, le c\u00f4t\u00e9 oppos\u00e9 \u00e0 l’angle droit s’appelle l’hypot\u00e9nuse<\/strong>. C’est le plus grand c\u00f4t\u00e9 du triangle.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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D\u00e9monstration :<\/strong><\/p>

\"\"<\/p>

Le quadrilat\u00e8re $MNOP$ est un losange car il a 4 c\u00f4t\u00e9s de m\u00eame longueur. De plus, la somme des mesures des deux angles aigus d’un triangle rectangle est \u00e9gale \u00e0 $90\u00b0$. Donc :
$\\widehat{\\text{DOP}}+\\widehat{\\text{DPO}}=\\widehat{\\text{DOP}}+\\widehat{\\text{CON}}
=90\u00b0$.
Ainsi, $\\widehat{\\text{PON}}=\\widehat{\\text{DOC}}-90\u00b0=180\u00b0-90\u00b0=90\u00b0$.
Le quadrilat\u00e8re $MNOP$ est un losange poss\u00e9dant un angle droit, c’est donc un carr\u00e9.<\/p>

Le quadrilat\u00e8re $ABCD$ est un carr\u00e9 car il a 4 angles droits et 4 c\u00f4t\u00e9 de m\u00eame longueur.<\/p>

On construit maintenant un quadrilat\u00e8re $EFGH$ en y repla\u00e7ant les 4 triangles rectangles.<\/p>

Le quadrilat\u00e8re $EFGH$ a 4 c\u00f4t\u00e9s de m\u00eame longueur donc c’est un losange. De plus, il poss\u00e8de un angle droit donc c’est un carr\u00e9.<\/p>

Les deux quadrilat\u00e8res $ABCD$ et $EFGH$ sont deux carr\u00e9s dont les c\u00f4t\u00e9s ont la m\u00eame longueur, ils ont donc la m\u00eame aire. Ainsi : $A_{\\text{ABCD}}=A_{\\text{EFGH}}$.<\/p>

En observant les deux carr\u00e9s, on en d\u00e9duit que l’aire du carr\u00e9 rouge est \u00e9gale \u00e0 la somme de l’aire du carr\u00e9 bleu et de l’aire du carr\u00e9 orange. Ainsi :
\\begin{eqnarray*}
A_{\\text{carr\u00e9 rouge}}&=&A_{\\text{carr\u00e9 bleu}}+A_{\\text{carr\u00e9 orange}}\\\\
\\text{c}^{2}&=&\\text{b}^{2}+\\text{a}^{2}
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Remarque :<\/strong> Ce th\u00e9or\u00e8me permet de calculer la longueur d\u2019un des c\u00f4t\u00e9s d\u2019un triangle rectangle connaissant les deux autres.<\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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C) Calculer une longueur d\u2019un c\u00f4t\u00e9 d\u2019un triangle rectangle<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Exemple :<\/strong> Le triangle $ABC$ ci-dessous est rectangle en $A$. Calculer la longueur $BC$.<\/p>

\"\"<\/p>

Le triangle $ABC$ est rectangle en $A$.
D’apr\u00e8s le th\u00e9or\u00e8me de Pythagore :
\\begin{eqnarray*}
\\text{BC}^{2}&=&\\text{AC}^{2}+\\text{AB}^{2}\\\\
\\text{BC}^{2}&=&4^{2}+3^{2}\\\\
\\text{BC}^{2}&=&16+9\\\\
\\text{BC}^{2}&=&25\\\\
\\text{BC}&=&\\sqrt{25}\\\\
\\text{BC}&=&5~\\text{cm}
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Exemple :<\/strong> Le triangle $DEF$ ci-dessous est rectangle en $E$. Calculer la longueur $EF$.<\/p>\n

\"\"<\/p>\n

Le triangle $DEF$ est rectangle en $E$.
D’apr\u00e8s le th\u00e9or\u00e8me de Pythagore :<\/p>\n

\\begin{eqnarray*}
\\text{DF}^{2}&=&\\text{ED}^{2}+\\text{EF}^{2}\\\\
8^{2}&=&3^{2}+\\text{EF}^{2}\\\\
64&=&9+\\text{EF}^{2}\\\\
\\text{EF}^{2}&=&64-9\\\\
\\text{EF}^{2}&=&55\\\\
\\text{EF}&=&\\sqrt{55}\\\\
\\text{EF}&\\approx &7,4~\\text{cm}
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Chapitre 2 Le th\u00e9or\u00e8me de Pythagore Chapitre 2 : Le th\u00e9or\u00e8me de Pythagore A) Racine carr\u00e9 d\u2019un nombre positif D\u00e9finition : $a$ d\u00e9signe un nombre positif. La racine carr\u00e9e de $a$ est le nombre positif dont le carr\u00e9 est $a$. Ce nombre est not\u00e9e $sqrt{a}$ (lire \u00ab\u00a0racine carr\u00e9e de $a$\u00a0\u00bb.) Ainsi, [sqrt{a}geq 0~~~et~~~left(sqrt{a}right)^{2}=a] Exemple : […]<\/p>\n","protected":false},"author":7440,"featured_media":14734,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"site-sidebar-layout":"default","site-content-layout":"page-builder","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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