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.
Answers
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.