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(C) Let the length of the line segment be $d$ and its direction cosines be $l, m,$ and $n$.The projections of the line segment on the $x, y,$ and $z$

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$5\sqrt{2}$

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(C) Let the length of the line segment be $d$ and its direction cosines be $l, m,$ and $n$.The projections of the line segment on the $x, y,$ and $z$ axes are given by $dl, dm,$ and $dn$ respectively.Given that $dl = 3, dm = 4,$ and $dn = 5$.Squaring and adding these equations,we get:$(dl)^2 + (dm)^2 + (dn)^2 = 3^2 + 4^2 + 5^2$$d^2(l^2 + m^2 + n^2) = 9 + 16 + 25$Since $l^2 + m^2 + n^2 = 1$ for any line segment,we have:$d^2(1) = 50$$d = \sqrt{50} = 5\sqrt{2}$.

If the projections of a line segment on the coordinate axes are $3, 4,$ and $5$,then its length is:

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