Prove that the diagonals of a rhombus are perpendicular bisectors of each other.
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°