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 Q1. In quadrilateral ABCD, the diagonals AC and BD meet at O. Suppose the four triangles AOB,BOC,COD and DOA are equal in area, prove that ABCD is a parallelogram. S1. [AOB] = [COD] => 1/2.OA.OB.Sin(AOB) = 1/2.OC.OD.Sin(COD) Angles AOB and COD are same. => OA.OB = OC.OD Similarly, OA.OD = OC.OB Multiply both => OA = OC and OB = OD => O bisects both diagonals. If the diagonals bisect each other, the quadrilateral is a ||gram. Why? Let the quadrilateral be A (0,0), B(x,0), C(a,b) D(p,q) O = Midpoint of AC = (a/2,b/2) = Midpoint of BD = ((p+x)/2,q/2) b/2 = q/2 => b = q => CD || AB and a = p+x Slope of AD = q/p Slope of BC = b/(a-x) = q/p => AD || BC Opposite sides are || => ABCD is a ||gram. H.P. Q2. In a parallelogram  A B C D A BC D , a point  P P on the segment  A B A B is taken such that  A P A B = A B A P ​ =   61 2022 2022 61 ​  and a point  Q Q on the segment  A D A D is taken such that  A Q A D = 61 2065...