Question

measures of all angles of the parallelogram.
(s. .
Diagonals of a parallelogram intersect each other at point O. If AO = 5, BO = 12
AB 13 then show that ABCD is a rhombus.​

Answer 1

AO = 5,

BO = 12,

AB = 13

[Given]  AO2 + BO2 = 52 + 122 = 25 + 144 

∴ AO2 + BO2 = 169 …..(i) 

AB2 = 132 = 169 ….(ii) 

∴ AB2 = AO2 + BO2

[From (i) and (ii)] 

∴ ∆AOB is a right-angled triangle. [Converse of Pythagoras theorem] 

∴ ∠AOB = 90° 

∴ seg AC ⊥ seg BD …..(iii) [A-O-C]

  ∴ In parallelogram ABCD, 

∴ seg AC ⊥ seg BD [From (iii)] 

∴ ABCD is a rhombus.