Page 226 - Ganit Kaushal

8.5 SQUARES AND RECTANGLES

Let us now consider figures that have four straight lines as their boundaries. We know that
these are called quadrilaterals.
If we make any closed curve with four straight lines as its boundary, it is called a quadrilateral
(see figures alongside).
What are squares and rectangles, then? They also have four straight lines as
their boundaries. What makes a quadrilateral a rectangle or a square?
Let’s recall the rules that make a quadrilateral a rectangle:
R1. The opposite sides are equal in length, and A B
R2. All the angles are 90°.

The rules for a square are as under:
S1. All four sides are equal in length, and O
S2. All the angles are 90°. D C
Consider this rectangle ABCD (see adjoining figure).

The points A, B, C and D are the ‘corners’ or ‘vertices’ of the rectangle. Lines AB, BC, CD and DA are sides
forming the ‘boundary’ of the rectangle. The blue sides AB and CD are called ‘opposite sides’, as they lie
opposite to each other. Likewise, the orange sides AD and BC are the other pair of opposite sides. The green
lines AC and BD are called ‘diagonals’. Their point of intersection, ‘O’ is called the ‘centre’ of the rectangle.

Its angles are ∠A, ∠B, ∠C, and ∠D, where ∠A = ∠B = ∠C = ∠D = 90°.
∠A and ∠C are called ‘opposite angles’ as they lie opposite to each other. Likewise, the angles ∠B and ∠D
form the other pair of opposite angles.
This rectangle can also be named in other ways — BCDA, CDAB, DABC, ADCB, DCBA, CBAD, and BADC.

Can a rectangle be named using any combination of the labels of its corners?
The answer is No! For example, it cannot be named ABDC or ACBD.
Can you tell why? Can you observe what names are allowed and what names are not?
Great! You are right. In a valid name, the corners occur in an order of travel around the rectangle
(clockwise or anticlockwise order), starting from any corner. So, the names like ABDC or ACBD are
not allowed because they neither follow the clockwise order nor the anticlockwise order.
Also, the corners, sides and names for squares are defined exactly in the same manner.

Rotated Squares and Rectangles
B
Take a rectangular piece of paper, where A B

AB = DC, and AD = BC (rule R1)
A
and ∠A = ∠B = ∠C = ∠D = 90° (rule R2)
C
It is rotated as shown in the figure.
D C
Is it still a rectangle? Can you answer? D
Wow, you are absolutely correct! It is still a rectangle because we can check that in the rotated piece
of paper, the opposite sides are still equal and all the angles are still 90°. Both the rules R1 and R2 are
satisfied. Therefore, a rectangle remains a rectangle even after it is rotated.

224 Mathematics-6