Question

angle abd =cbd=pqb =90.if ab =x units cd=y units pq=z units prove 1/x +1/y=1/z

Answer 1

see the diagram.

ΔABD and ΔPQD are similar, as the corresponding sides are parallel.
      x / z = BD / QD
             =>  1/x = QD /(z * BD)   ---(1)

ΔCDB and ΔPQB are similar, as corresponding sides are parallel.
     y / z = BD / BQ
             =>  1/y = BQ / (z * BD) ---(2)

Add the two equations: 
         1/x + 1/y = (BQ + QD) / (z * BD) = 1 / z