Question

Show that the diagonals of a square as equal and bisect each other at right angles

Answer 1

yes they intersect each oter at 90 degree because all the sides of square are equal

and the angles are of 90 degree

Answer 2

$\huge{\mathfrak{Solution:}}$



Let ABCD be a square. Let the diagonals AC and BD intersect each other at a point O. To prove

that the diagonals of a square are equal and bisect each other at right angles, we have to

prove AC = BD, OA = OC, OB = OD, and ∠AOB = 90º.

In ΔABC and ΔDCB,

AB = DC (Sides of a square are equal to each other)

∠ABC = ∠DCB (All interior angles are of 90)

BC = CB (Common side)

So, ΔABC ≅ ΔDCB (By SAS congruency)

Hence, AC = DB (By CPCT)

Hence, the diagonals of a square are equal in length.

In ΔAOB and ΔCOD,

∠AOB = ∠COD (Vertically opposite angles)

∠ABO = ∠CDO (Alternate interior angles)

AB = CD (Sides of a square are always equal)

So, ΔAOB ≅ ΔCOD (By AAS congruence rule)

Hence, AO = CO and OB = OD (By CPCT)

Hence, the diagonals of a square bisect each other.

In ΔAOB and ΔCOB,

As we had proved that diagonals bisect each other, therefore,

AO = CO

AB = CB (Sides of a square are equal)

BO = BO (Common)

So, ΔAOB ≅ ΔCOB (By SSS congruency)

Hence, ∠AOB = ∠COB (By CPCT)

However, ∠AOB + ∠COB = 1800 (Linear pair)

2∠AOB = 1800

∠AOB = 900

Hence, the diagonals of a square bisect each other at right angles.