In triangle $ABC$, we denote the midpoints of sides $CB$ and $CA$ by the letters $K$ and $L$. We know that the quadrilateral $ABKL$ has a perimeter of $10 \text{ cm}$ and triangle $KLC$ has a perimeter of $6 \text{ cm}$. Calculate the length of segment $KL$.

(J. Mazák)
Segment $K L$ is the midline of triangle $A B C$ parallel to $A B$, since $K$ and $L$ are the midpoints of sides $B C$ and $A C$. Therefore, $2|K L|=|A B|$ and also $|A L|=|L C|$ and $|C K|=|K B|$.

The perimeter of triangle $K L C$ is $|C K|+|K L|+|L C|=6$. The perimeter of quadrilateral $A B K L$ is

$$
|A B|+|B K|+|K L|+|L A|=10 \text {. }
$$

The sum on the left side of the aforementioned equality can be expressed according to previous observations as $2|K L|+(|C K|+|K L|+|L C|)$, or $2|K L|+6$. The mentioned equality thus takes the form

$$
2|K L|+6=10
$$

from which we get $2|K L|=4$, i.e., $|K L|=2(\mathrm{~cm})$.

Evaluation. 2 points for identifying and justifying that $|A B|=2|K L| ; 2$ points for considerations about perimeters; 2 points for calculating $|K L|$.