Question

Prove that if the diagonals of a quadrilateral bisect each other,
then it is a parallelogram.
A solid cube of side 12 cm is cut into 8 cubes of equal volume.
What will be the side of the new cube? Also find the ratio
between their lateral surface areas.​

Answer 1

ABCD is an quadrilateral with AC and BD are diagonals intersecting at O.

It is given that diagonals bisect each other.

∴ OA=OC and OB=OD

In △AOD and △COB

⇒ OA=OC [ Given ]

⇒ ∠AOD=∠COB [ Vertically opposite angles ]

⇒ OD=OB [ Given ]

⇒ △AOD≅△COB [ By SAS Congruence rule ]

∴ ∠OAD=∠OCB [ CPCT ] ----- ( 1 )

Similarly, we can prove

⇒ △AOB≅△COD

⇒ ∠ABO=∠CDO [ CPCT ] ---- ( 2 )

For lines AB and CD with transversal BD,

⇒ ∠ABO and ∠CDO are alternate angles and are equal.

∴ Lines are parallel i.e. AB∥CD

For lines AD and BC, with transversal AC,

⇒ ∠OAD and △OCB are alternate angles and are equal.

∴ Lines are parallel i.e. AD∥BC

Thus, in ABCD, both pairs of opposite sides are parallel.

∴ ABCD is a parallelogram.