Question

A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is
(a) 7
(b) $\sqrt{38}$
(c) $\sqrt{155}$
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{a}\right)7 \\ \mathrm{The}\mathrm{given}\mathrm{points}\left(2,3,5\right)\mathrm{and}\left(5,9,7\right)\mathrm{are}\mathrm{two}\mathrm{diagonally}\mathrm{opposite}\mathrm{vertices}\mathrm{of}\mathrm{the}\mathrm{parallelopiped}\mathrm{as}\mathrm{all}\mathrm{of}\mathrm{their}\mathrm{coordinates}\mathrm{are}\mathrm{different}. \\ \therefore \mathrm{Edges}\mathrm{of}\mathrm{the}\mathrm{parallelopiped}=\left|2-5\right|,\left|3-9\right|\mathrm{and}\left|5-7\right| \\ =3,6\mathrm{and}2 \\ \mathrm{Now}, \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{diagonal}\mathrm{of}\mathrm{the}\mathrm{parallelopiped}=\sqrt{{\left(3\right)}^2+{\left(6\right)}^2+{\left(2\right)}^2} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\sqrt{9+36+4} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\sqrt{49} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=7 \\ \mathrm{Hence},\mathrm{length}\mathrm{of}\mathrm{the}\mathrm{diagonal}\mathrm{of}\mathrm{the}\mathrm{parallelopiped}\mathrm{formed}\mathrm{by}\mathrm{the}\mathrm{planes}\mathrm{parallel}\mathrm{to}\mathrm{coordinate}\mathrm{planes}\mathrm{and}\mathrm{drawn}\mathrm{through}\mathrm{points}\left(2,3,5\right)\mathrm{and}\left(5,9,7\right)\mathrm{is}7\mathrm{units}. \end{gathered}$$