Consider the triangle shown below:
If
$x = \sqrt{c^2 - h^2}$, $y = \sqrt{b^2 - h^2}$ and $z = \sqrt{a^2 - h^2}$
Then, what will be the value of $h$ in terms of $a$, $b$ and $c$?
Equation of the plane is $x'/a + y'/b + z'/c = 1$ and the shortest distance from origin is $h$.
It is easier to use the following formula:
The distance of the point $(x_0, y_0, z_0)$ from the plane $ax + by + cz + d = 0$ is given by $\frac{|ax_0 + by_0+cz_0 + d|}{\sqrt{a^2+b^2+c^2}}$
Using this, the distance is $\frac{1}{\sqrt{\frac{1}{a^2}+ \frac{1}{b^2}+\frac{1}{c^2}}}$