Q4 of 138 Page 24

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.

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


More from this chapter

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2

Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

 3

(Pythagoras’s Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

 5

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

 6

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