In the figure, AP and BQ are perpendiculars to PQ and AP = BQ, prove that R is the mid-point of PQ and AB.

In ΔAPR and ΔBQR,
   AP = BQ ...... (Given)
∠ARP = ∠BRQ ...... (Vertically opposite angles)
∠APR = ∠BQR ...... (each 90°)
ΔAPR ≅ ΔBQR ...... (RHS Criterion)
 ∴PR = RQ ..... (c.p.c.t.)
and AR = RB ...... (c.p.c.t.) 
Hence R is the mid-point of AB and PQ.