Exercice 3
Soit A A A B B B m ∈ R m\in\R m ∈ R
Déterminer les valeurs de m m m ( A , m ) (A,m) ( A , m ) ( B , m − 1 ) (B,m-1) ( B , m − 1 ) G G G G A → \overrightarrow{GA} G A G B → \overrightarrow{GB} GB
Construire le point G G G m = 2 m=2 m = 2
Correction
G G G ( A , m ) (A,m) ( A , m ) ( B , m − 1 ) (B,m-1) ( B , m − 1 )
m + m − 1 ≠ 0 ⟺ m ≠ 1 2 m+m-1\ne0 \iff m\ne\dfrac12 m + m − 1 = 0 ⟺ m = 2 1
si : m ≠ 1 2 m\ne\dfrac12 m = 2 1
m G A → + ( m − 1 ) G B → = 0 ⃗ m\overrightarrow{GA}+(m-1)\overrightarrow{GB}=\vec{0} m G A + ( m − 1 ) GB = 0
Si m = 2 m=2 m = 2
2 G A → + ( 2 − 1 ) G B → = 0 ⃗ 2\overrightarrow{GA}+(2-1)\overrightarrow{GB}=\vec{0} 2 G A + ( 2 − 1 ) GB = 0 2 G A → + G A → + A B → = 0 ⃗ 2\overrightarrow{GA}+\overrightarrow{GA}+\overrightarrow{AB}=\vec{0} 2 G A + G A + A B = 0 A G → = 1 3 A B → \overrightarrow{AG}=\frac13\overrightarrow{AB} A G = 3 1 A B