## Task 2 - 200722

For a triangle, the following is required:

The lengths of the sides $a, b, c$ measured in cm should be natural numbers, the side length $a$ should be exactly $36\%$ of the perimeter $u$, and the side length $b$ should be exactly $48\%$ of the perimeter.

a) Investigate whether there is a triangle under these conditions with a perimeter $u=25 \mathrm{~cm}$! If this is the case, give its side lengths!

b) Investigate whether there is a triangle under the given conditions with a perimeter $u>25$ $\mathrm{cm}$! If this is the case, give its side lengths!

c) Determine all lengths $u$ that are less than $100 \mathrm{~cm}$ and can occur as the perimeter of a triangle whose side lengths meet the stated requirements! Determine the side lengths of such a triangle for each of these values $u$!
a) $36 \%$ of $25 \mathrm{~cm}$ is $36 \cdot \frac{25}{100} \mathrm{~cm}=9 \mathrm{~cm}$, and $48 \%$ of $25 \mathrm{~cm}$ is $12 \mathrm{~cm}$.

Furthermore, $25 \mathrm{~cm}-9 \mathrm{~cm}-12 \mathrm{~cm}=4 \mathrm{~cm}$. Since the triangle inequalities

$$
12 \mathrm{~cm}+4 \mathrm{~cm}>9 \mathrm{~cm}, \quad 4 \mathrm{~cm}+9 \mathrm{~cm}>12 \mathrm{~cm}, \quad 9 \mathrm{~cm}+12 \mathrm{~cm}>4 \mathrm{~cm}
$$

are satisfied, there exists a triangle with side lengths $a=9 \mathrm{~cm}, b=12 \mathrm{~cm}, c=4 \mathrm{~cm}$. This triangle meets the given requirements.

b), c) If a length $u=z \mathrm{~cm}$ appears as the perimeter of a triangle that meets the requirements of the problem, then: The number $z$ is a natural number, and $\frac{36}{100} z=\frac{9}{25} z$ is also a natural number, namely the measure of $a$.

Thus, $9z$ is divisible by 25. Since 9 is coprime with 25, $z$ must be divisible by 25. Therefore, only lengths $u=z \mathrm{~cm}$ with $z=25n$ for natural $n$ can appear as the perimeter of a triangle that meets the given requirements.

Due to the requirement $z<100$, only values $n<4$ are possible, i.e., the length specifications $u=25 \mathrm{~cm}, u=50 \mathrm{~cm}, u=75 \mathrm{~cm}$.

For each such perimeter specification: $36 \%$ of $25n$ is $9n$; $48 \%$ of $25n$ is $12n$; and $25n-9n-12n=4n$. Again, the triangle inequalities are satisfied, so there are triangles that meet the requirements of the problem for these perimeter specifications.

By substituting the values $1, 2, 3$ for $n$, we obtain the desired side lengths, namely for $n=1$ the values from part a) and for $n=2$ for the perimeter $u=50 \mathrm{~cm}$ the side lengths $a=18 \mathrm{~cm}$, $b=24 \mathrm{~cm}$, $c=8 \mathrm{~cm}$, and for $n=3$ for the perimeter $u=75 \mathrm{~cm}$ the side lengths $a=27 \mathrm{~cm}$, $b=36 \mathrm{~cm}$, $c=12 \mathrm{~cm}$.