Question

line segment joining the midpoint M and end of the parallel sides ab and CD respectively of trapezium ABCD is perpendicular to both the sides ab and BC prove that a b is equal to BC please help

Answer 1

Construction:− Join CM and DM.In ∆CMN and ∆DMNMN=MN (Common)∠CNM=∠DNM=90° (MN Is perpendicular to DC)CN=DN (Since N is the mid point of DC). By SAS congruency ∆CMN≅∆DMN Therefore,CM=DM (CPCT)∠CMN=∠DMN (CPCT)∠AMN=∠BMN=90 (Since MN is perpendicular to AB)
So,∠AMN−∠CMN=∠BMN−∠DMN (Since ∠CMN=∠DMN )∠AMD=∠BMC. In ∆AMD and ∆BMCDM=CM (Proved above)∠AMD=∠BMC (Proved above)AM=BM (M is the mid point of AB) By SAS congruency ∆AMD≅∆BMC. Therefore,AD=BC (CPCT)Hence Proved