Lesson Introduction to properties of a rectangle

In this lesson we are going to deal with rectangles and their basic properties. Further we are going to build a deep understanding of its properties and will prove them simultaneously.

Rectangle

The rectangle is one of the most commonly known quadrilaterals.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 are divided in their intersection point into two.


The rectangle has following special properties:

1.The rectangle is a type of parallelogram having opposite sides parallel and congruent.

2.The corner angles are all right angles (90 degrees).

3.Diagonals of a rectangle are equal and bisect each other.

4.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 .





Since opposite angles in a parallelogram are equal.

Hence, ........................................(1)


Now, AB || DC,

Hence, By Sum of angles property in a parallel lines,





.........................................(2)

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

.........................................(3)

From Equations (1),(2) and (3)



Hence all angle in a rectangle are equal to 90 degrees.

Proof 2. The opposite sides are equal in a Rectangle.

Rectangle is a special type of parallelogram whose all angles are 90 degrees. From the property of parallelogram the opposite sides of rectangle are hence equal.

Hence ............................................(4)

Proof 3. Diagonals of a rectangle are equal and bisect each other.

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

Triangles ABD and ADC are Right Angled Triangles right angled at Angle(DAC) and Angle(ADC) respectively.

From Pythagorean theorem,

In Triangle ABD,

.............................................(5)

In Triangle ADC,

.............................................(6)

From equations (4),(5) and (6)







Hence Diagonals are of equal length in a rectangle.

From property of parallelogram the diagonals are divided into at their point of intersection. As diagonals are equal in a rectangle hence,

Diagonals of a rectangle are equal and bisect each other.

Proof 4. 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 Angle(DAC) and Angle(ADC) respectively.

From Pythagorean theorem,

In Triangle ABD,

.............................................(7)

In Triangle ADC,

.............................................(8)

From equations (4),(7) and (8)





QED