#### 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

7

`sqrt(38)`

`sqrt(155)`

none of these

#### Solution

7

\[\text{ The given points } \left( 2, 3, 5 \right) \text{ and } \left( 5, 9, 7 \right) \text{ are two diagonally opposite vertices of the parallelopiped as all of their coordinates are different }. \]

\[ \therefore \text{ Edges of the parallelopiped } = \left| 2 - 5 \right|, \left| 3 - 9 \right| \text{ and } \left| 5 - 7 \right| \]

\[ = 3, 6 \text{ and } 2\]

\[\text { Now} , \]

\[\text{ Length of the diagonal of the parallelopiped } = \sqrt{\left( 3 \right)^2 + \left( 6 \right)^2 + \left( 2 \right)^2}\]

\[ \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} = \sqrt{9 + 36 + 4}\]

\[ \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} = \sqrt{49} \]

\[ \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} = 7\]

\[\text{ Hence, length of the diagonal of the parallelopiped formed by the planes parallel to coordinate planes and drawn through points } \left( 2, 3, 5 \right) \text { and } \left( 5, 9, 7 \right) \text{ is 7 units } . \]