Students define a rhombus, explore its properties and their proofs, and use these properties to solve problems
After completing this tutorial, you will be able to complete the following:
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 self-similarity 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.
The English word "rhombus" is derived from the Ancient Greek "rhombos", meaning "spinning top." The plural of rhombus can be either rhombi or rhombuses.
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:
Each pair consists of two congruent triangles, owing to the side-side-side congruency property of triangles. To see this, note that each triangle in a pair shares a side and that the nonshared sides are congruent by the definition of a rhombus.
In order to proceed, we recall some basic facts about isosceles triangles. First, consider the altitude of an isosceles triangle. The altitude of any triangle is perpendicular to the base. Additionally, the altitude of an isosceles triangle divides the triangle into smaller congruent triangles, and is therefore the perpendicular bisector of the base. This follows from the principle known as CPCTC, which stands for "corresponding parts of congruent triangles are congruent." The congruent portions of any isosceles triangle can therefore be marked as shown.
These facts apply to the congruent triangles in a rhombus, thereby allowing us to mark the rhombus diagram as below. By marking the rhombus in this way, we see that the diagonals are perpendicular bisectors of each other and that opposite angles are congruent. Additionally, by using additional facts about transversals and triangles, one can straightforwardly see that opposite sides are parallel and that two consecutive angles are supplementary. These facts are collected in the corollaries of the proposition below:
Proposition: A rhombus is divided into two congruent isosceles triangles by each of its diagonals.
Corollary 1: The diagonals are angle bisectors.
Proof: By SSS congruency, and similarly . This yields that line BC bisects angle ABD and ACD and line AD bisects angle BAC and BDC.
Corollary 2: The diagonals are perpendicular to each other.
Proof: If we consider triangle BCD, we see that since the sum of the interior angles of a triangle is 180 degrees. Dividing both sides by 2, we see that This means that
Corollary 3: The diagonals of a rhombus are bisectors of each other.
Proof: By SAS congruency , so Similarly , so
Corollary 4: Two consecutive angles in a rhombus are supplementary and opposite sides are parallel.
Proof: By considering , we see that Now, the measure of angle BDC is and the measure of angle DCA is so the sum of these two consecutive angles is 180 degrees; thus, they are supplementary. Using the same logic, we can show that any two consecutive angles are supplementary.
AD is a transversal for AB and CD. By the above, angles BAC and ACD are supplementary. By the converse of the consecutive interior angles theorem, we have that AB is parallel to CD. Similarly, we can show that BC is parallel to AD
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should know the definition of a quadrilateral; be able to identify the angle bisector, median, and diagonal of a quadrilateral; know the properties of an isosceles triangle; determine the interior and supplementary angles of an isosceles triangle; and remember the properties of the height of an isosceles triangle.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||Angle bisector, congruent side, interior angle opposite angle|