PQRS is a square with the diagonals PR and SQ. Points W and X are located respectively on the sides PQ and QR such that Angle XSW = 45^{o}.
V is the midpoint of WX while T and U are respectively the points of intersection of PR with the lines WS and XS.
Determine the length of VU, given that VT = 5.
When I can't figure out the Geometry, I go analytic. Usually this makes a huge mess, but here it's not so bad.
I put S at the origin
P=(1,0)
Q=(1,1)
S=(0,1)
PR is the line y=1x
X gets coordinates (1,a)
Line WS ends up having slope (a+1)/(1a)
W = ((1a)/(a+1),1)
being the midpoint of WX,
V = (1/(a+1),(1+a)/2)
the intersection points:
U=((1a)/2,(1+a)/2)
T=(1/(a+1),a/(a+1))
[conveniently, TV and UV are vertical and horizontal]
each distance works out to
(a^2+1)/(2a+2)
so TV = UV
if VT is given as 5 then VU is also 5
Edited on December 14, 2007, 4:11 pm

Posted by Jer
on 20071214 14:26:19 