## Task 3 - 110713

Günther draws a triangle $\triangle A B C$ and notices:

The numerical value of the perimeter $u$ of his triangle $\triangle A B C$, measured in centimeters, is a prime number. Furthermore, $B C=a=6 \mathrm{~cm}, A C=b=2 \mathrm{~cm}$.

Determine $A B=c$ and $u$!
From the triangle inequality, for the triangle $\triangle A B C: ca-b$. According to the problem, this implies $4 \mathrm{~cm}<c<8 \mathrm{~cm}$.

Since the measure of the perimeter $u$ is supposed to be a prime number and the measures of $a$ and $b$ are integers, the measure of $c$ must also be an integer $z$ such that $4<z<8$. Therefore, $z$ can only be 5, 6, or 7.

Let $z=5$. Then $u=a+b+c=13 \mathrm{~cm}$, and 13 is a prime number.

Let $z=6$. Then $u=14 \mathrm{~cm}$.

Let $z=7$. Then $u=15 \mathrm{~cm}$.

In the last two cases, the measure of $u$ is not a prime number. Therefore, $c=5 \mathrm{~cm}, u=13 \mathrm{~cm}$ is the only solution to the problem.