Answer :

Given:- Rhombus OABC i.e all sides are equal

To Prove:- Diagonals are perpendicular bisector of each other



Proof:- Let, O at the origin


D is the point of intersection of both diagonals


be position vector of A and C respectively


Then,




Now,




as AB = OC


……(i)


Similarly



……(ii)


Tip:- Directions are important as sign of vector get changed


Magnitude are same AC = OB = √a2 + c2


Hence from two equations, diagonals are equal


Now let’s find position vector of mid-point of OB and AC




and




Magnitude is same AD = DC = OD = DB = 0.5(√a2 + c2)


Thus the position of mid-point is same, and it is the bisecting point D


By Dot Product of OB and AC vectors we get,






As the side of a rhombus are equal OA = OC




Hence OB is perpendicular on AC


Thus diagonals of rhombus bisect each other at 90°


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