Question

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

Answer 1

Answer:

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Answer 2

Given : ABCD is a square.

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. ( opposite 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 )

Δ OAD ≅ Δ OCB. (ASA property)

OA = OC --------------------(1)

Similarly OB = OD -----------------(2)

From (1) and (2) AC and BD bisect each other.

Now in a Δ OBA and Δ ODA,

OB = OD ( from (2) )

BA = DA

OA = OA ( common line )

Δ AOB = Δ AOD-------------------(3) ( by CPCT )

∠AOB + ∠AOD = 180°. (linear pair)

2∠AOB = 180°

∠AOB = ∠AOD =90°

∴AC and BD bisect each other at right angles.

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