1. In the right-angled triangle $A B C$, the median dropped from the right angle to the hypotenuse has a length of 1. If it is known that one of the angles formed by this median and the hypotenuse is $60^{\circ}$, determine the perimeter and area of triangle $A B C$.
Solution. Let $M$ be the midpoint of the hypotenuse $AB$. Then $\overline{AM}=\overline{BM}=\overline{CM}=1$ and $\overline{AB}=\overline{AM}+\overline{MB}=2$. Let $\angle AMC=60^{\circ}$. Then triangle $\triangle AMC$ is equilateral, so we get $\overline{AC}=1$. From the Pythagorean 

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theorem for $ABC$ we have $\overline{BC}=\sqrt{3}$. Therefore, the perimeter of $ABC$ is $L=3+\sqrt{3}$, and the area is $P=\frac{\sqrt{3}}{2}$.