9.1. Can the length of one of the medians of a right triangle be a) $51\%$ of the length of the hypotenuse? b) $49\%$ of the length of the hypotenuse?
Answer. a) Can; b) cannot. Solution. It is clear that the median is drawn to the midpoint of one of the legs, otherwise its length would be equal to half the hypotenuse. Let $\triangle A B C$ be a right triangle with a right angle at $C$ and $B C=a, A C=b$. Without loss of generality, we can assume that the median $A M$ is drawn to the midpoint $M$ of the leg $C B . \quad$ Let $k=A M / A B$. Then we have $k \cdot \sqrt{a^{2}+b^{2}}=\sqrt{b^{2}+\frac{a^{2}}{4}}, \quad$ i.e., $\left(4 k^{2}-1\right) a^{2}=4\left(1-k^{2}\right) b^{2}$. If $\frac{1}{2}<k<1$, then from here we will obtain the desired ratio of the legs. If, however, $0<k<\frac{1}{2}$, then in the obtained relation the left and right sides have different signs.