JAMB 1985 · UME · Q41

$GHIJKLMN$ is a cube of side $a$ , and $HN$ is a diagonal joining opposite corners. Find the length of $HN$ .

Worked solution (try it first)

First cross one face: the face diagonal has length $\sqrt{a^2 + a^2} = a\sqrt2$ .
$HN$ is the hypotenuse of a right-angled triangle with that face diagonal and a vertical edge $a$ : $HN^2 = 2a^2 + a^2 = 3a^2$ .
So $HN = a\sqrt3$ , option E.