Question

If the projection of a line segment on $x,y$ and $z$ axes are respectively $3,4$ and $5$, then the length of the line segment is

• A. $3\sqrt{2}$
• B. $5\sqrt{2}$
• C. $6\sqrt{2}$
• D. None of these

Answer 1

The correct option is B $5\sqrt{2}$

Let $l,m,n$ be the d.c's of the given line segment $PQ.$

$\therefore l=\cos \alpha ,m=\cos \beta ,n=\cos \gamma$

where $\alpha ,\beta ,\gamma$ are the angles which the line segment $PQ$ makes with the axes.

Suppose length of line segment $PQ=r.$

Thus, projection of line segment $PQ$ on x-axis $=PQ\cos \alpha ,=rl.$

Also the projection of line segment $PQ$ on x-axis $=3$ (given).

$\therefore lr=3$

Similarly, $mr=4,nr=5.$

Now squaring and adding there equations, we get

${\left(lr\right)}^2+{\left(mr\right)}^2+{\left(nr\right)}^2=r^2[3^2+4^2+5^3]$

$\Rightarrow r^2(l^2+m^2+n^2)=9+16+25$

$\Rightarrow r^2=50[\because l^2+m^2+n^2=1]$

$\Rightarrow r=\sqrt{50}=5\sqrt{2}$