The Diagonals of a Parallelogram Bisect Each Other


 
 
Concept Explanation
 

The Diagonals of a Parallelogram Bisect Each Other

Theorem 4: The diagonals of a parallelogram bisect each other.

GIVEN   A parallelogram ABCD such that its disgonal AC and BD intersect at O.

To Prove OA =  OC and OB = OD

Proof  Since ABCD is a parallelogram. Therefore,

      AB large parallel DC    and   AD large parallel BC

Now, AB large parallel DC and transversal AC intersects them at A and C respectively.

large therefore    <BAC = DCA                      [ large because  Alternate interior angles are equal ]

large Rightarrow   <BAO = <DCO                                  .....(i)

Again, AB large parallel DC and BD intersectsthem at B and D respectively.

large therefore    <ABD = <CDB                   [ large because Alternate interior angles are equal ]

large Rightarrow   <ABO = <CDO                                .....(ii)

Now, in large Delta s AOB and COD, we have

      <BAO = DCO                      [From (i)]

          AB = CD                        [large because Opposite sides of a large parallelgm are equal]

and, <ABO = <CDO                   [From (ii)]

So, by ASA congruence criterion

     large DeltaAOB large cong large DeltaCOD

large Rightarrow  OA = OC and OB = OD

Hence, OA = OC and OB = OD

Converse Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
Given: A quadrilateral ABCD in which the diagonals AC and BD intersect at O such that they bisect each other that is OA = OC and OB = OD

To Prove: The Quadrilateral ABCD is a Parallelogram

Proof: In Delta AOB and Delta COD

        OA = OC        [Given ]

angle AOB = angle COD      [Vertically Opposite Angles]

       OB = OD       [Given]

Therfore Delta AOB cong Delta COD     [ SAS Criteria of Congruence]

  angle OAB = angle OCD            [C.P.C.T.]

But they are alternate interior angles when AB and CD are straight lines and AC is the transversal

As they are equal AB || CD

Similarly we can prove that AD || BC

As both the opposite pair of sides are parallel to each other.

Hence ABCD is a parallelogram

Illustration: In a parallelogram ABCD the diagonals AC and BD intersect at O, AC = 12.6 cm and BD = 5.8 cm. Find the length of OA, OB, OC and OD.

Solution: ABCD is a parallelogram and diagonals of the parallelogram bisect each other.

O is the midpoint of AC

OA=frac{1}{2}AC=frac{1}{2} X 12.6 = 6.3 cm

OC = OA = 6.3 cm

Similarly O is the mid point of BD

OB=frac{1}{2}BD=frac{1}{2} X 5.8 = 2.9 cm

OD = OB = 2.9 cm

Hence OA = 6.3 cm, OB = 2.9cm, OC = 6.3 cm and OD = 2.9cm

 

 
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