Q. 84.1( 435 Votes )

ABCD is a rectangle in which diagonal AC bisects A as well as C Show that:

(i) ABCD is a square

(ii) Diagonal BD bisects B as well as D

Answer :


Given: ABCD is a rectangle, 

A = C

A = C
To Prove: ABCD is a square
Proof:

DAC = DCA (AC bisects A and C)

CD = DA (Sides opposite to equal angles are also equal)


However,


DA = BC and AB = CD (Opposite sides of a rectangle are equal)

AB = BC = CD = DA


ABCD is a rectangle and all of its sides are equal.

Hence, ABCD is a square

Hence, Proved.

(ii) Let us join BD

In ΔBCD,


BC = CD (Sides of a square are equal to each other)


CDB = CBD (Angles opposite to equal sides are equal)


However,


CDB = ABD (Alternate interior angles for AB || CD)


CBD = ABD


BD bisects B


Also,


CBD = ADB (Alternate interior angles for BC || AD)


CDB = ABD


BD bisects D

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