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Géométrie

Cours : Géométrie > Chapitre 9 

Leçon 3: Le théorème de Pythagore et la distance entre deux points dans le plan repéré
Mathématiques
Géométrie
Le théorème de Pythagore
Le théorème de Pythagore et la distance entre deux points dans le plan repéré
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Distance entre deux points

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Suivant les pays les notations sont différentes. Nous utilisons AB pour désigner la longueur du segment [AB]. Peut-être utilisez-vous |AB|.
Quels que soient A, left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis et B, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis la start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd entre A et B, c'est-à-dire la longueur du segment open bracket, A, B, close bracket est :
A, B, equals, square root of, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared, end square root
D'où vient cette formule et comment l'appliquer ?

Comprendre la formule

On place les points de coordonnées left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, space, ;, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis et left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, space, ;, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis.
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two.
Il s'agit de calculer la start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd entre ces deux points, c'es-à-dire la longueur du segment tracé en bleu.
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points.
Pour calculer cette start color #11accd, start text, l, o, n, g, u, e, u, r, end text, end color #11accd, on trace un triangle rectangle ce qui permettra d'utiliser le théorème de Pythagore.
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle.
La longueur du côté de l'angle droit tracée en vert est start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle. The hypotenuse of the right triangle is unknown and the side made from the point at x one, y one and x two, y one is labeled x two minus x one.
De même, la longueur de l'autre côté de l'angle droit est start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10 :
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle. The hypotenuse of the right triangle is unknown and the side made from the point at x one, y one and x two, y one is labeled x two minus x one. The third side is labeled y two minus y one.
On utilise le théorème de Pythagore :
start color #11accd, question mark, end color #11accd, squared, equals, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared
La start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd cherchée est :
start color #11accd, question mark, end color #11accd, equals, square root of, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared, end square root
On obtient la formule de la distance entre deux points de coordonnées données.
Si vous n'arrivez pas à mémoriser cette formule, il est toujours possible de tracer un triangle rectangle dont l'hypoténuse est le segment en question et d'appliquer le théorème de Pythagore.

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