Thalès : triangles emboîtés

Le plan - Mathématiques 4e

Exercice 1 : Démontrer l'égalité de deux quotients avec le théorème de Thalès (étapes très guidées)

Soit ABC un triangle.
Soit D un point de la droite (AB) et E un point de la droite (AC) tels que (BC) // (DE)

Démontrer \(\frac{AB}{AD} = \frac{AC}{AE}\).

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Exercice 2 : Théorème des milieux, un cercle

On considère le cercle suivant de centre O et de rayon 8.

Sachant que \(RS = 7\), que vaut \(OT\) ?

Exercice 3 : Application du théorème de Thalès sur 2 paires de triangles emboîtés adjacents (plus dur)

Soit la figure suivante :

\(E\), \(G\), \(C\) sont alignés, \(E\), \(H\), \(F\) sont alignés, \(E\), \(I\), \(D\) sont alignés
\((GH)\) \(//\) \((CF)\)
\((HI)\) \(//\) \((FD)\)
\( EG = 9 \), \( GC = 10 \), \( EH = 9 \),

Calculer le quotient \( \dfrac{GH}{CF} \).
On donnera la réponse sous la forme d'un entier ou d'une fraction simplifiée.

En déduire la longueur \( HF \).
On donnera la réponse sous la forme d'un entier ou d'une fraction simplifiée.

Exercice 4 : Application du théorème de Thalès sur un triangle particulier

Soit la figure suivante :

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\(H\), \(O\), \(I\) sont alignés, \(J\), \(K\), \(I\) sont alignés
\((OK)\) \(//\) \((HJ)\)
\(IK = 3\) et \(IO = 2\)

Calculer \( IJ \).
On donnera la réponse sous la forme d'un entier ou d'une fraction simplifiée.

Calculer \( HI \).
On donnera la réponse sous la forme d'un entier ou d'une fraction simplifiée.

Exercice 5 : théorème des milieux, deux cercles, centres non confondus

On considère la figure suivante dans laquelle \(O_{1}\) et \(O_{2}\) sont les centres des cercles représentés :

Sachant que \(O_{1}O_{2} = 8\), que vaut \(LM\) ?

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