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.

The freedom of choosing where W lies along PQ results in a constrained choice of where X lies along QR.  As XR decreases with any increase in PW, XR can be made equal to PW and stay in conformity with the statement of the problem. This presents a symmetry in which VU has the same length as VT, so in this case the length of VU is also 5.

Assuming that there is only one answer, that answer must be 5.

Moving W along in Geometer's Sketchpad, with the stated constraints followed by the software, shows that VU is indeed always equal to VT, though I don't have a mathematical proof.

Posted by Charlie on 2007-12-13 13:05:36