Lesson Introduction to properties of a rectangle

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A rectangle is one of the most commonly known  quadrilaterals.
In this lesson we are going to deal with rectangles and their basic properties.  Further we are going to build a deep understanding of their properties and will prove them simultaneously.

Rectangle

A parallelogram in which each angle is 90 degrees is called a rectangle.  Hence a rectangle has all the properties of a parallelogram.


    


The properties of a Parallelogram common to rectangle are:

1.  Opposite sides of a parallelogram are equal.

2.  Opposite angles of a parallelogram are equal.

3.  Diagonals of a parallelogram bisect each other in their intersection point.


The rectangle has following special properties:

1.  The corner angles all are right angles (90°).

2.  Diagonals of a rectangle are equal.

3.  A square of a diagonal length is equal to a sum of squares of its sides lengths.


Proof of properties of a Rectangle


Proof 1.  If in a parallelogram one angle is 90 degrees then all angles are 90 degrees.
Consider a parallelogram  ABCD,  where it is given that  LBCD = 90°.





Since opposite angles in a parallelogram are equal, we have

 LDAB =  LBCD = 90°.                                             (1)


Now,  AB || DC.

Hence,  by the  Sum of angles  property in a parallel lines,

 LDAB +  LADC=180°,

90° +  LADC = 180°,

 LADC = 180°-90° = 90°.                                        (2)

Again,  by the property of parallelogram that opposite angles are equal,

 LABC =  LADC = 90°.                                             (3)

From equations  (1),  (2)  and  (3)

 LBCD =  LDAB =  LADC =  LABC = 90°.

Hence,  all  angles  in a rectangle are equal to 90 degrees.


Proof 2.  Diagonals of a rectangle are equal.

Consider two  Triangles   ABD   and   ADC  containing two diagonals  BD  and  AC  respectively.

Triangles  ABD  and  ADC  are Right Angled Triangles right angled at  LDAC  and  LADC  respectively.

From Pythagorean theorem,

In Triangle  ABD,

BD%5E2=%28AB%5E2%2BAD%5E2%29.                                                    (4)

In Triangle  ADC,

AC%5E2=AD%5E2%2BDC%5E2.                                                       (5)

From equations  (5)  and  (6)

BD%5E2=AC%5E2=%28AD%5E2%2BDC%5E2%29=%28AB%5E2%2BAD%5E2%29,

BD%5E2=AC%5E2,

BD=AC.

Hence,  diagonals of a rectangle are of equal length.

From the property of a parallelogram the diagonals of a rectangle bisect each other at their intersection point. Thus  diagonals of a rectangle are of equal length and bisect each other.


Proof 3.  A square of a diagonal length is equal to a sum of squares of its sides lengths.

Consider two Triangles  ABD  and  ADC  containing two diagonals BD and AC respectively.

Triangles  ABD  and  ADC  are  Right Angled Triangles  right angled at  LDAC  and  LADC  respectively.

From  Pythagorean theorem,

In Triangle  ABD,

BD%5E2=%28AB%5E2%2BAD%5E2%29.                                                    (7)

In Triangle  ADC,

AC%5E2=AD%5E2%2BDC%5E2.                                                       (8)

From equations  (7)  and   (8)

BD%5E2=AC%5E2=%28AD%5E2%2BDC%5E2%29=%28AB%5E2%2BAD%5E2%29,

BD%5E2=AC%5E2.

QED

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