Parallelogram Family

A Parallelogram has many properties.

1. Opposite sides are parallel & equal

2. Both pairs of opposing angles are equal.

3. This means the 2 angles at both the ends of any side would be supplementary i.e total to 180 degrees.

4. Each diagonal divides the parallelogram into 2 congruent triangles

5. Both diagonals bisect each other.

It is very easy to prove that if all the mid points of the 4 sides of any quadrilateral are joined, they form a parallelogram.

These properties have been used to construct mechanical links for transmitting & magnifying motion in machines. They are used in a class of linkages called Pantographs.

In electric trains & trams, pantographs help the electrical motor to maintain continuous contact with the overhead electric cable in spite of constant up & down movement. Pantographs are used in engineering drawing instruments for drawing accurate parallel lines. They are also used in reproducing diagrams to scale. In houses they are used to hold mirrors which can be adjusted to any height and length.

Rhombus - If in a Parallelogram, adjacent sides are equal (which would mean all 4 sides are equal) we get a Rhombus.

If in a Kite, the opposing sides are equal, then also we get a Rhombus. Hence a Rhombus is related to the quadrilateral family in two ways; one through the Parallelogram and the second through the Kite.

Rectangle - If any one of the angles of a Parallelogram is a right angle, then we get a Rectangle. (In a rectangle, all angles are equal to right angles).

Square - If the adjacent sides of a Rectangle are equal then we get a Square.

On the other hand, if any one of the angles in a Rhombus is a right angle, then we get a Square.

Chapter 23.10 show the hierarchical relation between different types of quadrilaterals. For example, all rectangles are parallelograms, but all parallelograms are not rectangles!

Rectangles & Squares are some of the first shapes that children are able to identify along with the circle. This is possibly because most of the objects in their immediate environment are of these shapes.

Since all the angles of the rectangle and square are right angles, the length of the diagonals can be calculated using the Pythagoras Theorem. In both the diagonals bisect each other.

Similarity of Rectangles

The property of similarity is applicable to rectangles also. Students normally learn about similarity only in relation to triangles and remember that in similar triangles, their corresponding angles are equal. Hence there is a common misconception that all rectangles are similar as all their angles are equal.

In case of rectangles, similarity depends on the equality of the ratio between the length & breadth. Visually, if 2 similar rectangles are aligned so that 2 of their adjacent sides and the vertex coincide, then their diagonals will be collinear.

Square – All squares are similar. This is the same as in case of circles. The square is the basis of the unit of measuring areas. Hence the units are called “square units”.

Rectangle & the Golden Ratio

The Greek letter ϕ, pronounced phi, is a number very much like π. Greeks called it the “golden ratio”. They considered a rectangle with ϕ as the ratio of its length & breadth, to be most pleasing to the eye.

Measurement of area

The principle of finding the area of any plane figure is to discover or construct a rectangle with the same area. Any rectangle can be subdivided into squares.

The area of a circle can be equated to that of a rectangle with one side equal to the circumference & the other equal to half the radius. A = (2πr) * (r/2) which simplifies to.

Construction & Tiling

Rectangular & Square shapes are easy to manufacture. Hence most constructions use rectangular & square shapes. They also cover an area without leaving any gaps. Hence most of the tiles which cover the floors of our houses would be rectangular or square in shape.

Tessellation

The art of covering a large plane surface with tiles of various shapes is called tessellation. It is both an art and an architectural science. Triangles, squares, rectangles and hexagons can tessellate by themselves. Certain other shapes can tessellate in combination with other shapes.

Tessellations with different shapes can be tried out as an exercise in math & art.