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RHOMBUS AND ITS PROPERTIES
Teacher Guide
Background Information
Rhombuses are a type of quadrilateral, or four-sided polygon, such that all four of its sides
are congruent. Rhombuses have been studied since before the time of Euclid, who
employed compass/ruler techniques to prove a variety of interesting theorems about
rhombuses. More recently, the mathematician Robert Penrose discovered that
rhombuses can be used to create remarkable tilings of the plane, now called Penrose
tilings. A Penrose tiling is a special tiling of the plane, chiefly characterized by selfsimilarity and a lack of translational invariance, meaning that no two shifts of the tiling
look the same and that any portion of the tiling looks similar to some larger portion.
Penrose tilings have important applications to quantum physics, number theory, and
geometry. Interestingly, rhombuses are only one of the three figures Penrose used to
create these tilings.
(Image credit: Wikipedia)
The English word “rhombus” is derived from the Ancient Greek “rhombos,” meaning
“spinning top.” The plural of rhombus can be either rhombi or rhombuses.
Examples:
Rhombuses enjoy a number of interesting properties, the most important of which follow
immediately from basic theorems about triangles. These properties are sometimes used
incorrectly to define rhombuses, most frequently by stating that they are parallelograms
with four congruent sides. This extra hypothesis is completely unnecessary; the fact that
any rhombus is a parallelogram follows from the congruency of its sides. Such things can
be discovered by identifying two pairs of isosceles triangles in the rhombus. Recall that an
isosceles triangle has two congruent sides. The isosceles triangles in a rhombus are shown
in the figure below:
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