Question

prove that the perpendicular drown from the vertex of a regular pentagon to the opposite side bisect the side

Answer 1

ABCDE be a regular pentagon.  AE be base. C be the top vertex from which a perpendicular is drawn to AE meeting AE at F.  So AFC and CFE are right angle triangles.  AC and EC are same as it is a regular pentagon. In these two triangles two sides are equal AC = EC and CE =CE. Angle CFA and CFE = 90.  So they rare congruent. AF = EF.