PARALLELOGRAM LAW

PARALLELOGRAM LAW

A parallelogram. The sides are shown in blue and the diagonals in red.

In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. Using the notation in the diagram on the right, the sides are (AB), (BC), (CD), (DA). But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, or (AB) = (CD) and (BC) = (DA), the law can be stated as,

${\displaystyle 2(AB)^{2}+2(BC)^{2}=(AC)^{2}+(BD)^{2}\,}$

If the parallelogram is a rectangle, the two diagonals are of equal lengths (AC) = (BD) so,

${\displaystyle 2(AB)^{2}+2(BC)^{2}=2(AC)^{2}\,}$

and the statement reduces to the Pythagorean theorem. For the general quadrilateral with four sides not necessarily equal,

${\displaystyle (AB)^{2}+(BC)^{2}+(CD)^{2}+(DA)^{2}=(AC)^{2}+(BD)^{2}+4x^{2},\,}$

where x is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that, for a parallelogram, x = 0, and the general formula simplifies to the parallelogram law.

Parallelogram law of forces

If two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant isrepresented in magnitude and direction by the diagonal passing through the point.

Explanation

Consider two forces Vector P and Vector Q acting at a point O inclined at an angle θ as shown in Fig..

The forces Vector P and Vector Q are represented in magnitude and direction by the sides OA and OB of a parallelogram OACB as shown in Fig.

The resultant Vector R of the forces Vector P and Vector Q is the diagonal OC of the parallelogram. The magnitude of the resultant is

R = root[ P2 +Q2 + 2PQcos θ ]

The direction of the resultant is α = tan-1[ Qsin θ  /  P+Qcos θ ]