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 Square And Intersection (Posted on 2007-12-13)
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 = 45o.

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.

 See The Solution Submitted by K Sengupta Rating: 2.5000 (2 votes)

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 Analytic | Comment 2 of 3 |
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=1-x

X gets coordinates (1,a)
Line WS ends up having slope (a+1)/(1-a)
W = ((1-a)/(a+1),1)
being the midpoint of WX,
V = (1/(a+1),(1+a)/2)

the intersection points:
U=((1-a)/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 2007-12-14 14:26:19

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