Origin-to-plane distance with a Lagrange multiplier

How far is the plane $2x+3y+4z=12$ from the origin?

$12/\sqrt{29}\approx2.228$. Minimise $x^2+y^2+z^2$ subject to the constraint: the Lagrange condition makes $(x,y,z)$ parallel to the normal $(2,3,4)$, so the nearest point is $\tfrac{12}{29}(2,3,4)$, at distance $12/\sqrt{4+9+16}$.