Notes
parallelogram law

Skip the Navigation Links | Home Page | Feeds

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
Views: Print | TeX | Source | Linked from: five semi-circles solution, five semi-circles techniques, four squares vii solution