582. Find the height and area of an equilateral triangle with side $a$.
In an isosceles triangle, the height coincides with the median (and the bisector). Therefore, it divides the equilateral triangle into two right triangles, in which the hypotenuse is $a$, one of the legs is $a / 2$, and the height $h$ is the other leg. Thus, it can be found using the formula:

$$
h=\sqrt{a^{2}-(a / 2)^{2}}=\sqrt{\frac{3}{4} a^{2}}=\frac{\sqrt{3}}{2} a
$$

Accordingly, the area of the equilateral triangle is $\frac{\sqrt{3}}{4} a^{2}$.

In a right triangle with legs 3 and 4, the hypotenuse, found using the Pythagorean theorem, turns out to be an integer: it is equal to $\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=\sqrt{25}=5$.

This, of course, does not always happen: by randomly taking two integer legs $a$ and $b$, we will most likely not get a perfect square ($a^{2}+b^{2}$ will not be the square of an integer). For example, for a triangle with legs 3 and 5, we get $3^{2}+5^{2}=9+25=34$, and the hypotenuse $\sqrt{34}$ will not be an integer. However, the example of a right triangle with integer sides that we provided is not the only one.