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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 |
| Answer» <html><body><p>:-ABCD is a Parallelogram.Diagonals AC & <a href="https://interviewquestions.tuteehub.com/tag/bd-389836" style="font-weight:bold;" target="_blank" title="Click to know more about BD">BD</a> bisects at O.p and q are the mid points of DO & BO <a href="https://interviewquestions.tuteehub.com/tag/respectively-1186938" style="font-weight:bold;" target="_blank" title="Click to know more about RESPECTIVELY">RESPECTIVELY</a>.◘ Required To Prove:-APCQ is a Parallelogram.◘ Proof:-By The Point of TrisectionAs, The diagonals of a Parallelogram Intrest each other, OB = ODAnd p & q are midpoints of OD & BO respectively. Now, Consider P & QBQ = PQ = DPNow,In ∆ADP & ∆BCQ→ AD = BC [Opposite Sides of a Parallelogram] → DP = BQ [Proved Above]→ ∠ADP = ∠CBQ [Alternative Interior <a href="https://interviewquestions.tuteehub.com/tag/angles-378243" style="font-weight:bold;" target="_blank" title="Click to know more about ANGLES">ANGLES</a>]∆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 <a href="https://interviewquestions.tuteehub.com/tag/might-560522" style="font-weight:bold;" target="_blank" title="Click to know more about MIGHT">MIGHT</a> be a Rectangle or Parallelogram.Now, Consider OB & OD➝ OB = OD➝ BQ+OQ = OP+DP➝ BQ+OQ = OP+BQ [.:. BQ = DP]➝ OQ = OPIn Quadrilateral APCQOQ = OPOA = OCSince, the Diagonals of the given it is definitely a Parallelogram.A Quadrilateral in which two pairs of <a href="https://interviewquestions.tuteehub.com/tag/opposites-1137374" style="font-weight:bold;" target="_blank" title="Click to know more about OPPOSITES">OPPOSITES</a> sides are equal and diagonals bisect each other then it is a Parallelogram.Hence, APCQ is a Parallelogram.@MrSovereign ツHope This Helps!!</p></body></html> | |