3.4.1. Rhombus

We obtain a Rhombus (which, you will see, is a parallelogram) as a special case of kite (which is not a parallelogram).

A Rhombus is a type of quadrilateral that can be understood as a special case of a kite. To visualize this, think about the kite you made earlier. A kite is defined as having two pairs of adjacent sides that are equal in length. However, when we make the lengths of the sides equal (AB = BC), the kite transforms into a rhombus. Also, it’s important to know that while all rhombuses are parallelograms, not all kites fit this definition.

Imagine a kite that you fly on a windy day. If you make sure that not only the two pairs of sides are equal but also make all four sides of equal length, you create a diamond shape, which is a rhombus.

A rhombus is a quadrilateral with sides of equal length. Since the opposite sides of a rhombus have the same length, it is also a parallelogram.

The defining feature of a rhombus is that all its sides are equal in length. This means that if you measure each side with a ruler, they all will give you the same measurement. Additionally, since a rhombus has opposite sides that are equal, it satisfies the definition of a parallelogram, which also has equal opposite sides. Because of these properties, a rhombus inherits all properties of parallelograms as well.

Think of a piece of jewelry shaped like a diamond. Every edge is of the same length, which makes it a rhombus. If you were to pull the corners of the diamond into a longer shape while still keeping those edges equal, you would create a parallelogram.

Property: The diagonals of a rhombus are perpendicular bisectors of one another.

The diagonals of a rhombus have a unique property. When you draw the diagonal lines from one corner of the rhombus to the opposite corner, these diagonals intersect each other at right angles (90 degrees). Furthermore, they bisect each other, meaning they cut each other in half at the point of intersection. This property is what helps in identifying a rhombus in geometric problems.

Imagine a square piece of pizza cut into four triangles. The lines you've drawn from one corner of the square to the opposite corner would represent the diagonals. These lines cross at the center of the pizza, dividing each slice into two equal halves and forming right angles.

Here is an outline justifying this property using logical steps. ABCD is a rhombus (Fig 3.29). Therefore it is a parallelogram too. Since diagonals bisect each other, OA = OC and OB = OD.

To explain why the diagonals intersect at right angles, consider a rhombus labeled ABCD. Since it is a parallelogram, its diagonals bisect each other at a point O. This means that the segments AO and OC are equal lengths, and similarly for BO and OD. Since we establish that triangles AOD and COD are congruent using the SSS (Side-Side-Side) criterion, we can conclude that the angles formed at the intersection are right angles.

Think of a large cross on a road. No matter how you measure each arm of the cross, they will meet at the center, dividing into four equal sections. This represents the property of the diagonals bisecting each other at right angles.

Example 7: RICE is a rhombus (Fig 3.30). Find x, y, z. Justify your findings.

In this example, we're given a rhombus named RICE, where certain measurements relate to the sides and diagonals. By using properties of equal lengths and angles within a rhombus, we can set up equations to find the values of x, y, and z. This practical application consolidates understanding by requiring the use of various properties, such as the fact that all sides are equal.

Consider a situation where you have a group of friends and you want to know how many of them are pairing up to play a game. If you can determine that each pair is equivalent to a side of a rhombus, just like finding out how many ways you can connect these points to play, you can utilize the same principles to calculate the number of friends.

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