In triangle $ABC$, we draw the altitude $AA_{1}$ from vertex $A$. What is the length of $BA_{1}$ if $AB=8 \mathrm{~cm}, AC=10 \mathrm{~cm}$, $BC=12 \mathrm{~cm}$.
If $B A_{1}=x$, then

$$
8^{2}-x^{2}=10^{2}-(12-x)^{2}
$$

from which

$$
x=4.5 \text{ cm}
$$

(Stolzer Imre, Györr.)