## Problem 4

Let $A B C D$ be a rectangle, with side lengths $A B=4 \mathrm{~cm}$ and $A D=3 \mathrm{~cm}$, and $\{O\}=A C \cap B D$.

Determine the length of the vector $\vec{v}=\overrightarrow{A B}+\overrightarrow{A O}+\overrightarrow{A D}$.

## Note

- Effective working time 3 hours.
- All problems are mandatory.
- 7 points are awarded for each correctly solved problem.


## National Mathematics Olympiad  Local stage -14.02.2015  9th Grade $\mathbf{M}_{2}$  Solutions and marking schemes
## Problem 4

We apply the Pythagorean theorem in $\triangle A B C \Rightarrow A C=5 \mathrm{~cm}$. $1 p$

We apply the parallelogram rule and $\overrightarrow{A B}+\overrightarrow{A D}=\overrightarrow{A C}$

$\vec{v}=\overrightarrow{A B}+\overrightarrow{A O}+\overrightarrow{A D}=\overrightarrow{A C}+\overrightarrow{A O}=\overrightarrow{A C}+\frac{1}{2} \overrightarrow{A C}=\frac{3}{2} \overrightarrow{A C}$

$|\vec{v}|=\left|\frac{3}{2} \overrightarrow{A C}\right|=\frac{3}{2} A C=\frac{15}{2} \mathrm{~cm}$.

$2 p$