Show that diagonal of a square are equal and bisect each other at righ

1.	Show that diagonal of a square are equal and bisect each other at right angle
Answer» Consider a square ABCD whose diagonals AC and BD intersect each other at a point O.We have to prove that AC = BD, OA = OC, OB = OD, and ∠AOB = 90º.Consider ΔABC and ΔDCB,AB = DC∠ABC = ∠DCB = 90°BC = CB (Common side)∴ ΔABC ≅ ΔDCB (By SAS congruency)∴ AC = DB (By CPCT)Hence, the diagonals of the square ABCD are equal in length.Now, consider ΔAOB and ΔCOD,∠AOB = ∠COD (Vertically opposite angles)∠ABO = ∠CDO (Alternate interior angles)AB = CD (Sides of a square are equal)∴ ΔAOB ≅ ΔCOD (By AAS congruence rule)∴ AO = CO and OB = OD (By CPCT)Hence, the diagonals of the square bisect each other.Similarly, in ΔAOB and ΔCOB,AO = COAB = CBBO = BO∴ ΔAOB ≅ ΔCOB (By SSS congruency)∴ ∠AOB = ∠COB (By CPCT)∠AOB + ∠COB = 180º (Linear pair)or, 2∠AOB = 180ºor, ∠AOB = 90ºHence, the diagonals of a square bisect each other at right angles.

Show that diagonal of a square are equal and bisect each other at right angle