Question

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

Answer 1

consider a square ABCD
diagonals are AC,and BD

To prove : AC = BD and AC and BD bisect each other at right angles.

Proof:

(i) In a Δ ABC and Δ BAD,

AB = AB ( common line)

BC = AD ( opppsite sides of a square)

∠ABC = ∠BAD ( = 90° )

Δ ABC ≅ Δ BAD ( By SAS property)

AC = BD ( by CPCT).

(ii) In a Δ OAD and Δ OCB,

AD = CB ( opposite sides of a square)

∠OAD = ∠OCB ( transversal AC )

∠ODA = ∠OBC ( transversal BD )

ΔOAD ≅ ΔOCB (ASA property)

OA = OC ---------(i)

Similarly OB = OD ----------(ii)

From (i) and (ii) AC and BD bisect each other.

Now in a ΔOBA and ΔODA,

OB = OD ( from (ii) )

BA = DA

OA = OA ( common line )

ΔAOB = ΔAOD ----(iii) ( by CPCT

∠AOB + ∠AOD = 180° (linear pair)

2∠AOB = 180°

∠AOB = ∠AOD = 90°

∴AC and BD bisect each other at right angles.