The Diagonals of a Parallelogram Bisect Each Other
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 Now, AB
Again, AB
Now, in <BAO = DCO [From (i)] AB = CD [ and, <ABO = <CDO [From (ii)] So, by ASA congruence criterion
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 OA = OC [Given ]
OB = OD [Given] Therfore 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 OC = OA = 6.3 cm Similarly O is the mid point of BD 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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