Notes
parallelogram law

Parallelogram Law

The parallelogram law relates the lengths of the diagonals of a parallelogram to the lengths of its sides.

ParallelogramLaw.png?

Specifically, as in the above diagram let aa and bb be the two side lengths of a parallelogram, and cc and dd the lengths of its diagonals. Then

c 2+d 2=2a 2+2b 2 c^2 + d^2 = 2 a^2 + 2 b^2

Proof from the Cosine Rule

ParallelogramLawProof.png?

The parallelogram law can be deduced from the cosine rule by considering two triangles inside the parallelogram. Since the two angles add up to 180 ∘180^\circ, we have cos(D)=−cos(C)\cos(D) = - \cos(C) and so from the cosine rule:

c 2 =a 2+b 2−2abcos(C) d 2 =a 2+b 2−2abcos(D)=a 2+b 2+2abcos(C) c 2+d 2=2a 2+2b 2 \begin{aligned} c^2 &= a^2 + b^2 - 2 a b \cos(C) \\ d^2 &= a^2 + b^2 - 2 a b \cos(D) = a^2 + b^2 + 2 a b \cos(C) \\ c^2 + d^2 = 2 a^2 + 2 b^2 \end{aligned}

Proof by Similar Triangles

ParallelogramLawProofByTriangles.png?

Using the two triangles formed by cutting the parallelogram along each of its diagonals it is possible to form a new parallelogram. The top edge of this parallelogram has length c 2+d 2c^2 + d^2 while the bottom edge has length a 2+b 2+a 2+b 2=2a 2+2b 2a^2 + b^2 + a^2 + b^2 = 2 a^2 + 2 b^2.

Created on June 24, 2021 07:18:23 by Andrew Stacey?