1. problems on plane

The base PQRS of a cube with each edge of length 2a is horizontal and V is the centre of the face opposite PQRS . Calculate the length(call this PX) of perpendicular from P to VQ .

This is what i did :

I found $VQ=a\sqrt{6}$ , PX will be its half .

$
PV=VQ=a\sqrt{6}
$

$PX=\sqrt{(a\sqrt{6})^2-(\frac{a\sqrt{6}}{2})^2}$

$=\frac{3\sqrt{2}}{2}a$

i got it wrong . where is my mistake ?

2. Originally Posted by thereddevils
The base PQRS of a cube with each edge of length 2a is horizontal and V is the centre of the face opposite PQRS . Calculate the length(call this PX) of perpendicular from P to VQ .

This is what i did :

I found $VQ=a\sqrt{6}$ , PX will be its half.

$
PV=VQ=a\sqrt{6}
$

$PX=\sqrt{(a\sqrt{6})^2-(\frac{a\sqrt{6}}{2})^2}$

$=\frac{3\sqrt{2}}{2}a$

i got it wrong . where is my mistake ?
The mistake is in red. X is not the midpoint of VQ. In fact, the sides of the isosceles triangle PVQ are $PV=VQ=a\sqrt6$ and $PQ=2a$. If Y is the midpoint of PQ then VY is perpendicular to PQ, from which you can use Pythagoras' theorem to find that $VY = a\sqrt5$. Now calculate the area of the triangle in two ways: it is $\tfrac12PQ.VY$ and also $\tfrac12VQ.PX$. From that, you can find PX.