SUBIECTUL III

Let point O be the midpoint of a straight segment $[A B]$. On the ray (OA, consider a point $\mathrm{E}$ such that $B E=5 \cdot A E$. Find the length of the segment $A B$ knowing that $E O=6 \mathrm{~cm}$.
## SUBJECT III

Let point $O$ be the midpoint of a straight line segment $[A B]$. On the ray (OA, consider a point $\mathrm{E}$ such that $B E=5 \cdot A E$. Find the length of the segment $\mathrm{AB}$, knowing that $\mathrm{EO}=6 \mathrm{~cm}$.
![](https://cdn.mathpix.com/cropped/2024_06_07_b5b01a68e98bf7ed8ee5g-3.jpg?height=366&width=970&top_left_y=884&top_left_x=725)

The problem has two cases:

$1 p$

CASE I: $\quad E \in(O A)$

| SOLUTION | SCORE |
| :--- | :---: |
| From $B E=5 \cdot A E$ and $A B=A E+B E$ it follows that $A B=6 \cdot A E$. | $1 \mathrm{p}$ |
| Since point O is the midpoint of segment $[A B]$, it follows that $O A=3 \cdot A E$ | $1 \mathrm{p}$ |
| $\Rightarrow A E=E O: 2=3 \mathrm{~cm} \Rightarrow A B=18 \mathrm{~cm}$. | $1 \mathrm{p}$ |

CASE II: $E \in(O A-[O A]$

| SOLUTION | SCORE |
| :--- | :---: |
| From $B E=5 \cdot A E$ and $A B=B E-A E \Rightarrow A B=4 \cdot A E \cdot$ | $1 \mathrm{p}$ |
| Since point O is the midpoint of segment $[A B] \Rightarrow O A=2 \cdot A E$ | $1 \mathrm{p}$ |
| $\qquad A E=E O: 3=2 \mathrm{~cm} \Rightarrow A B=8 \mathrm{~cm}$ | $1 \mathrm{p}$ |