What is the length of VW from this following cube?

Question

Cube ABCD.EFGH with edge a cm.

P,Q,R,S,T is in the midpoint of AB,CD,EH,AD,and EG.

U at the midpoint of PQ.

UV is projection of UT at EPQH and RW is projection of RS at EPQH.

What is the length of VW?

Johan · Accepted Answer

Answer: VW = 4*a*sqrt(5)/5

Calculation:
If I understand correctly, U is at the center of ABCD, and T is at the center of EFGH.

So we have a rectangle SUTR, with UT = SR = a, and SU = TR = a/2.

Use Pythagoras to calculate RU = a*sqrt(5)/2.

Notice that triangle VRT is congruent with triangle TRU, because angle RVT is a right angle, and angle VRT = angle TRU.

Because of the angle VRT = angle TRU, we can determine that

VR:RT = TR:RU ==> 
VR/RT = RT/RU ==> multiply left and right by RT
VR = RT*RT/RU

RT = TR = a/2 and RU = a*sqrt(5)/2 (as already calculated), so

VR = (a/2)*(a/2)/(a*sqrt(5)/2) = (a/2)/sqrt(5) = a/(2*sqrt(5))

multiply nominator and denominator by sqrt(5):

VR = a*sqrt(5)/10

Likewise, because triangle WUS is congruent with SUR, we can determine that WU = a*sqrt(5)/10 = VR.

Now we can determine VW:

VW = RU - VR - WU 
= a*sqrt(5) - a*sqrt(5)/10 - a*sqrt(5)/10 
= 8*a*sqrt(5)/10
= 4*a*sqrt(5)/5

? · Answer

Have you written this correctly?  U is the same as point S.

Let&#39;squestion · Answer

I shall try later.