the parallelogram ABCD AC and BD are its diagonals intersect at p is the midpoint of deo and q is mid point of OB prove that a p c q is a parallelogram
Answers
◘ Given:-
- ABCD is a Parallelogram.
- Diagonals AC & BD bisects at O.
- p and q are the mid points of DO & BO respectively.
◘ Required To Prove:-
- APCQ is a Parallelogram.
◘ Proof:-
By The Point of Trisection
As, The diagonals of a Parallelogram Intrest each other, OB = OD
And p & q are midpoints of OD & BO respectively.
Now, Consider P & Q
BQ = PQ = DP
Now,
In ∆ADP & ∆BCQ
→ AD = BC [Opposite Sides of a Parallelogram]
→ DP = BQ [Proved Above]
→ ∠ADP = ∠CBQ [Alternative Interior Angles]
∆ADP ≅ ∆BCQ [By SAS axiom]
- AP = CQ [By CPCT]
Now,
In ∆DCP & ∆BAQ
→ DC = BA [Opposite Sides of Parallelogram]
→ DP = BQ [Proved Above]
→ ∠CDP = ∠ABQ [Alternative Interior Angles]
∆CDP ≅ ∆ABQ [By SAS axiom]
- AQ = CP [By CPCT]
As, The opposite sides are equal it might be a Rectangle or Parallelogram.
Now, Consider OB & OD
➝ OB = OD
➝ BQ+OQ = OP+DP
➝ BQ+OQ = OP+BQ [.:. BQ = DP]
➝ OQ = OP
In Quadrilateral APCQ
- OQ = OP
- OA = OC
Since, the Diagonals of the given it is definitely a Parallelogram.
A Quadrilateral in which two pairs of opposites sides are equal and diagonals bisect each other then it is a Parallelogram.
- Hence, APCQ is a Parallelogram.
@MrSovereign ツ
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Hope This Helps!!
