Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Question

Philoid · Accepted Answer

Given:- Parallelogram OABC

To Prove:- AC² + OB² = OA² + AB² + BC² + CO²

Proof:- Let, O at origin

Therefore,

Distance/length of AC

By triangular law:-

the the vectors form sides of triangle

⇒

As AB = OC and BC = OA

From figure

⇒

⇒ ……(i)

Similarly, again from figure

⇒

⇒ ……(ii)

Now,

Adding equation (i) and (ii)

⇒ ……(iii)

Take RHS

OA² + AB² + BC² + CO²

……(iv)

Thus from equation (iii) and (iv), we get

LHS = RHS

Hence proved