A point P lies in the first octant of a coordinate system measured in metres…

A point $P$ lies in the first octant of a coordinate system measured in metres. An engineer knows only its distances from the three axes. The distance of $P$ from the $x$-axis is $\sqrt{13}\ \text{m}$. Its distance from the $y$-axis is $\sqrt{20}\ \text{m}$. Its distance from the $z$-axis is $5\ \text{m}$. The engineer wants the coordinates of $P$. Let $P = (a, b, c)$. Write the three distances as three equations in $a^2$, $b^2$ and $c^2$. Add them to find $a^2 + b^2 + c^2$, and then find $a$, $b$ and $c$. Give the square of the distance of $P$ from the origin, which is a whole number. All three coordinates are positive.