Question

show that the diagonals of a regular polygon are equal

Answer 1

Let us consider a polygon (pentagon) ABCDE
As all the diagonals are equal AC = BD = CE = AD = BE

Solution:
we know that ∠A = ∠B = ∠C = ∠D = ∠E = 108 degree
Let us consider ∆ABC and ∆ADE
Here,AB = AE                                    
BC = DE                                     
∠ABC = ∠AED = 108 degree            
so, ∆ABC = ∆ADE                         

and AC = AD         -----(1) 

Similarly ∆BCD = ∆CDE
BD = CE               -------- (2)
 
∆ ABC = ∆BCD             
 AC = BD                  -------- (3)
 
 ∆ABC = ∆ABE
 BE = AC   -------- (4)
 
Equation the equations (1), (2), (3) and (4)
AC = BD = AD = CE = BE
Thus all the diagonals of a regular polygon are equal

Answer 2

Given: Consider a regular pentagon ABCDE……..
To prove: All the diagonals of the pentagon are equal………
 i.e. AC = AD = BD = CE = BE……………
   Proof:.…..
ABCDE is a regular pentagon………
Þ AB = BC = CD = DE = AE and ∠A = ∠B = ∠C = ∠D = ∠E = 108°.………
  Consider ∆ABC and ∆ADE……………
AB = AE ...... (Given)……..
BC = DE ….......... (Given)..........
∠ABC = ∠AED = 108° .............. (Given)............
∴ ∆ABC ≅ ∆ADE ..........
                 (SAS congruency rule)..........
 ∴ AC = AD .... (corresponding sides of congruent triangles) -------- (1).....
  Similarly ∆BCD ≅ ∆CDE.......
⇒ BD = CE -------- (2)......
   ∆ ABC ≅ ∆BCD ..........
   ⇒ AC = BD -------- (3)...........
    ∆ABC ≅ ∆ABE............
 ⇒ BE = AC -------- (4)...........
  From equations (1), (2), (3) and (4), we get....
AC = BD = AD = CE = BE...........
  Therefore, all the diagonals of a pentagon are equal.…………