3.4. Some Special Parallelograms

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

DO THIS Recall the paper-cut kite you made earlier.

Kite-cut Rhombus-cut When you cut along ABC and opened up, you got a kite. Here lengths AB and BC were different. If you draw AB = BC, then the kite you obtain is called a rhombus. Note that the sides of rhombus are all of same length; this is not the case with the kite.

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. So, a rhombus has all the properties of a parallelogram and also that of a kite. Try to list them out. You can then verify your list with the check list summarised in the book elsewhere.

A rhombus is a special type of quadrilateral where all four sides are of equal length. Unlike a kite, where only two pairs of adjacent sides are equal, a rhombus has all sides equal. Since opposite sides are equal and parallel, a rhombus falls under the broader category of parallelograms. Thus, it shares properties such as having opposite angles that are equal and diagonals that bisect each other.

Imagine looking at a diamond ring. The shape of the diamond, which resembles a rhombus, has all its sides equal in length. This is why when the ring is viewed from different angles, it maintains that symmetrical shape, just like a rhombus.

The most useful property of a rhombus is that of its diagonals.

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

DO THIS Take a copy of rhombus. By paper-folding verify if the point of intersection is the mid-point of each diagonal. You may also check if they intersect at right angles, using the corner of a set-square.

Here is an outline justifying this property using logical steps. ABCD is a rhombus. Therefore it is a parallelogram too. Since diagonals bisect each other, OA = OC and OB = OD. We have to show that m∠AOD = m∠COD = 90°.

In a rhombus, the diagonals have a unique property where they not only bisect each other but are also perpendicular to each other. This means that when the diagonals intersect, they form right angles (90 degrees). To demonstrate this, one can use a paper folding technique to physically visualize how the diagonals meet at their midpoints and intersect at right angles.

Think of a creased paper. When you fold a square or rhombus diagonally, the two creases will meet at the center and create four right angles. This interaction is reminiscent of how the diagonals of a rhombus interact — the intersection point creates a clear, symmetrical shape.

A rectangle is a parallelogram with equal angles. What is the full meaning of this definition? Discuss with your friends.

If the rectangle is to be equiangular, what could be the measure of each angle? Let the measure of each angle be x°. Then 4x° =360° (Why)? Therefore, x° =90° Thus each angle of a rectangle is a right angle. So, a rectangle is a parallelogram in which every angle is a right angle. Being a parallelogram, the rectangle has opposite sides of equal length and its diagonals bisect each other.

A rectangle is defined as a special type of parallelogram where all angles are equal to 90 degrees. This means that not only are the opposite sides of equal length, but every internal angle conforms to being a right angle. While all rectangles are parallelograms, not all parallelograms are rectangles, as rectangles have the additional condition of having equal angles.

Think about a book lying flat on a table. The corners of the book form right angles, much like the angles in a rectangle. This consistent right-angle characteristic is what makes the book's shape a rectangle — paralleling a flat, perfect structure.

In a parallelogram, the diagonals can be of different lengths. (Check this); but surprisingly the rectangle (being a special case) has diagonals of equal length.

Property: The diagonals of a rectangle are of equal length.

The diagonals of a rectangle, unlike those in a general parallelogram, are always of equal length. This property can be established via congruency of triangles formed by the diagonals. By breaking the rectangle into two triangles, it can be shown that these triangles are congruent, leading to the conclusion that the diagonals must also be equal.

Envision a rectangular photo frame. When you measure the lengths of the diagonals from corner to corner, both measurements will be the same. This mirrors how the property of equal diagonals operates in rectangles.

A square is a rectangle with equal sides. This means a square has all the properties of a rectangle with an additional requirement that all the sides have equal length. The square, like the rectangle, has diagonals of equal length.

A square is a unique form of a rectangle that has both equal-length sides and right angles. It combines all the features of rectangles and rhombuses into one shape. The diagonals of the square not only meet the criteria of being equal in length but are also perpendicular to each other. Because of this, the square holds properties of all previous quadrilaterals discussed: parallelogram, rhombus, and rectangle.

Imagine a chessboard, where every square on the board is equal in size. This symmetry keeps the board organized and makes every piece move consistently, like the properties associated with a square in geometry.

In a square the diagonals.

(i) bisect one another (square being a parallelogram) (ii) are of equal length (square being a rectangle) and (iii) are perpendicular to one another.

Hence, we get the following property.

Property: The diagonals of a square are perpendicular bisectors of each other.

The properties of a square's diagonals are unique because they bisect each other at right angles and are of equal length. These characteristics allow the square to maintain its symmetry and balance. When you fold a square from corner to corner, the diagonal lines will create an 'X' shape where they intersect at the center, and each diagonal divides the square into equal halves.

Think of a kite flying in the sky. If you were to draw the strings connecting the top of the kite to each corner, they would form an 'X' shape that looks just like the diagonals of a square — meeting at the center and forming right angles.

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