J

\colorbox{orange}{Démontrer\ si\ un\ triangle\ est\ rectangle\ ou\ non.}

Nouvel exercice

Voir/cacher la rédaction

C

A

Si le triangle n'apparaît pas entièrement, 
il est possible de déplacer les points colorés.

t_1: {3, 4, 5}
type = Random(0, 1)
t_2: {5, 12, 13}
t_3: {8, 15, 17}
t_4: {7, 24, 25}
t_5: {20, 21, 29}
t_6: {12, 35, 37}
t_7: {9, 40, 41}
t_8: {28, 45, 53}
t_9: {11, 60, 61}
t_{10}: {16, 63, 65}
t_{11}: {33, 56, 65}
t_{12}: {48, 55, 73}
t_{13}: {13, 84, 85}
t_{14}: {36, 77, 85}
t_{15}: {39, 80, 89}
t_{16}: {65, 72, 97}
triplets: {t_1, t_2, t_3, t_4, t_5, t_6, t_7, t_8, t_9, t_{10}, t_{11}, t_{12}, t_{13}, t_{14}, t_{15}, t_{16}}
triplet = RandomElement(triplets)
k = Random(1, 5)
cotés: {1, 2, 3}
coté1 = RandomElement(cotés)
cotés_2: cotés \ {coté1}
coté2 = RandomElement(cotés_2)
cotés_3: cotés_2 \ {coté2}
coté3 = Element(cotés_3, 1)
Alphabet: {"A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X", "Y", "Z"}
np1 = RandomElement(Alphabet)
A = (-2.700038214502712, 1.8060178794780397)
numtriplet = Random(1, 16)
b = If(type ≟ 1, (Element[triplet, coté2] * k / 10), (Element[triplet, coté2] * k / 10) + Random[1, 9] / 10)
c = If(type ≟ 1, (Element[triplet, coté3] * k / 10), (Element[triplet, coté3] * k / 10) + Random[1, 9] / 10)
a = If(type ≟ 1, (Element[triplet, coté1] * k / 10), (Element[triplet, coté1] * k / 10) + Random[1, 9] / 10)
lgueurs: {a, b, c}
lg1 = Element(lgueurs, 1)
lg2 = Element(lgueurs, 2)
lg3 = Element(lgueurs, 3)
lg_{max} = Max(lgueurs)
pos_{max} = IndexOf(lg_{max}, lgueurs)
carré1: lg1^(2)
carré2: lg2^(2)
carré3: lg3^(2)
carré12: carré1 + carré2
carré13: carré1 + carré3
carré23: carré3 + carré2
coté_1: {coté1}
correction = true
B = Point(Circle[A, lg3])
f = Segment(A, B)
d = Circle(A, lg2)
e = Circle(B, lg1)
C = Intersect(d, e, 1)
g = Segment(B, C)
h = Segment(C, A)
Alphabet2: Alphabet \ {np1}
np2 = RandomElement(Alphabet2)
depart: false
Alphabet3: Alphabet2 \ {np2}
np3 = RandomElement(Alphabet3)
D = (-12.933741475794196, 8.061681001090673)
E = (-1.015221214168137, 8.17088210296421)
texte3: "\text{Le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est tel que " + (LaTeX[np1]) + "" + (LaTeX[np2]) + " = " + (LaTeX[lg3]) + " ; " + (LaTeX[np1]) + "" + (LaTeX[np3]) + " = " + (LaTeX[lg2]) + " et " + (LaTeX[np2]) + "" + (LaTeX[np3]) + " = " + (LaTeX[lg1]) + ".
\\Ce triangle est-il rectangle ?}"
texte4: "\text{Si le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle,}\\
\text{il l'est en le point " + (LaTeX[np1]) + " car le segment [" + (LaTeX[np2]) + "" + (LaTeX[np3]) + "] est le côté du triangle de plus grande longueur.}\\" + (LaTeX[np2]) + "" + (LaTeX[np3]) + "² = " + (LaTeX[lg1]) + "^2 = " + (LaTeX[carré1]) + " \\" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² + " + (LaTeX[np1]) + "" + (LaTeX[np3]) + "² = " + (LaTeX[lg3]) + "^2 + " + (LaTeX[lg2]) + "^2 = " + (LaTeX[carré3]) + " + " + (LaTeX[carré2]) + " = " + (LaTeX[carré23]) + "\\
\text{Je constate que } " + (LaTeX[np2]) + "" + (LaTeX[np3]) + "²=" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "²+" + (LaTeX[np1]) + "" + (LaTeX[np3]) + "²\\
 \text{donc, d'après la réciproque du théorème de Pythagore,\\
\underline{le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle en " + (LaTeX[np1]) + "}.}"
texte5: "\text{Si le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle,}\\
\text{il l'est en le point " + (LaTeX[np2]) + " car le segment [" + (LaTeX[np1]) + "" + (LaTeX[np3]) + "] est le côté du triangle de plus grande longueur.}\\" + (LaTeX[np1]) + "" + (LaTeX[np3]) + "² = " + (LaTeX[lg2]) + " ^2 =  " + (LaTeX[carré2]) + " \\" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² + " + (LaTeX[np2]) + "" + (LaTeX[np3]) + "² =  " + (LaTeX[lg3]) + "^2 +  " + (LaTeX[lg1]) + "^2 =  " + (LaTeX[carré3]) + " + " + (LaTeX[carré1]) + " =  " + (LaTeX[carré13]) + "\\
\text{Je constate que } " + (LaTeX[np1]) + "" + (LaTeX[np3]) + "²=" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "²+" + (LaTeX[np2]) + "" + (LaTeX[np3]) + "²\\
 \text{donc, d'après la réciproque du théorème de Pythagore,\\
\underline{le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle en " + (LaTeX[np2]) + "}.}"
texte6: "\text{Si le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle,}\\
\text{il l'est en le point " + (LaTeX[np3]) + " car le segment [" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "] est le côté du triangle de plus grande longueur.}\\" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² = " + (LaTeX[lg3]) + " ^2 =  " + (LaTeX[carré3]) + " \\" + (LaTeX[np1]) + "" + (LaTeX[np3]) + "² + " + (LaTeX[np2]) + "" + (LaTeX[np3]) + "² = " + (LaTeX[lg2]) + " ^2 + " + (LaTeX[lg1]) + " ^2 =  " + (LaTeX[carré2]) + " + " + (LaTeX[carré1]) + "  = " + (LaTeX[carré12]) + " \\
\text{Je constate que } " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "²=" + (LaTeX[np1]) + "" + (LaTeX[np3]) + "²+" + (LaTeX[np2]) + "" + (LaTeX[np3]) + "²\\
 \text{donc, d'après la réciproque du théorème de Pythagore,\\
\underline{le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle en " + (LaTeX[np3]) + "}.}"
texte7: "\text{Si le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle,}\\
\text{il l'est en le point " + (LaTeX[np1]) + " car le segment [" + (LaTeX[np2]) + "" + (LaTeX[np3]) + "] est le côté du triangle de plus grande longueur.}\\" + (LaTeX[np2]) + "" + (LaTeX[np3]) + "² =  " + (LaTeX[lg1]) + "^2 =  " + (LaTeX[carré1]) + " \\" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² + " + (LaTeX[np1]) + "" + (LaTeX[np3]) + "² = " + (LaTeX[lg3]) + " ^2 +  " + (LaTeX[lg2]) + "^2 = " + (LaTeX[carré3]) + "  + " + (LaTeX[carré2]) + "  = " + (LaTeX[carré23]) + " \\
\text{Je constate que  " + (LaTeX[np2]) + "" + (LaTeX[np3]) + "² }≠ \text{ " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² + " + (LaTeX[np1]) + "" + (LaTeX[np3]) + "²}\\
 \text{donc, d'après la contraposée du théorème de Pythagore,\\
\underline{le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " n'est pas rectangle}.}"
texte8: "\text{Si le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle,}\\
\text{il l'est en le point " + (LaTeX[np2]) + " car le segment [" + (LaTeX[np1]) + "" + (LaTeX[np3]) + "] est le côté du triangle de plus grande longueur.}\\
\text{" + (LaTeX[np1]) + "" + (LaTeX[np3]) + "² }=  " + (LaTeX[lg2]) + " ^2 =  " + (LaTeX[carré2]) + "  \\
\text{" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² + " + (LaTeX[np2]) + "" + (LaTeX[np3]) + "² } =  " + (LaTeX[lg3]) + " ^2 + " + (LaTeX[lg1]) + "  ^2 = " + (LaTeX[carré3]) + "  + " + (LaTeX[carré1]) + "   = " + (LaTeX[carré13]) + "  \\
\text{Je constate que  " + (LaTeX[np1]) + "" + (LaTeX[np3]) + "² }≠ \text{ " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² + " + (LaTeX[np2]) + "" + (LaTeX[np3]) + "²}\\
 \text{donc, d'après la contraposée du théorème de Pythagore,\\
\underline{le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " n'est pas rectangle}.}"
texte9: "\text{Si le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " est rectangle,}\\
\text{il l'est en le point " + (LaTeX[np3]) + " car le segment [" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "] est le côté du triangle de plus grande longueur.}\\" + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² =  " + (LaTeX[lg3]) + " ^2 =  " + (LaTeX[carré3]) + "  \\" + (LaTeX[np1]) + "" + (LaTeX[np3]) + "² + " + (LaTeX[np2]) + "" + (LaTeX[np3]) + "² = " + (LaTeX[lg2]) + "  ^2 + " + (LaTeX[lg1]) + "  ^2 =  " + (LaTeX[carré2]) + "  +  " + (LaTeX[carré1]) + "  = " + (LaTeX[carré12]) + "  \\
\text{Je constate que  " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "² }≠ \text{ " + (LaTeX[np1]) + "" + (LaTeX[np3]) + "² + " + (LaTeX[np2]) + "" + (LaTeX[np3]) + "²}\\
 \text{donc, d'après la contraposée du théorème de Pythagore,\\
\underline{le triangle " + (LaTeX[np1]) + "" + (LaTeX[np2]) + "" + (LaTeX[np3]) + " n'est pas rectangle}.}"