Question

In a figure angle ABD= Angle CDB=Angle PQB= 90 degree if AB=x units,CD=y units and PQ= Z units,prove that 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)

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

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

Answer 2

ΔABD and ΔPQD are similar, as the corresponding sides are parallel.

      x / z = BD / QD   =>  1/x = QD /(z * BD)

ΔCDB and ΔPQB are similar, as corresponding sides are parallel.

     y / z = BD / BQ   =>  1/y = BQ / (z * BD)

Add the two equations: 

         1/x + 1/y = (BQ + QD) / (z * BD) = 1 / z