Lesson Introduction to properties of a rectangle

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A rectangle is one of the most commonly known &nbsp;<A HREF=http://en.wikipedia.org/wiki/Quadrilateral>quadrilaterals</A>.

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

<A HREF=http://www.algebra.com/algebra/homework/Rectangles/Rectangle.wikipedia>Rectangle</A>

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

&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 160, 144, -10, 10, -10, 8, 
      line( -6,   -6,   4, -6) , 
      line( -6,   -6,  -6,  1.5), 
      line( -6,    1.5, 4,  1.5), 
      line(  4,    1.5, 4, -6),
      line( -6,   -6,   4,  1.5),
      line(  4,   -6,  -6,  1.5),
    locate( -6.5, -6.5, A),
    locate(  4.5, -6.5, B),
    locate(  4.5,  2.5, C),
    locate( -7,    3,   D)
)}}}

The properties of a <A HREF=http://www.algebra.com/algebra/homework/Rectangles/Parallelogram.wikipedia>Parallelogram</A> common to rectangle are:

1. &nbsp;Opposite sides of a parallelogram are equal.

2. &nbsp;Opposite angles of a parallelogram are equal.

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

The rectangle has following special properties:

1. &nbsp;The corner angles all are right angles (90�).

2. &nbsp;Diagonals of a rectangle are equal.

3. &nbsp;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. &nbsp;If in a parallelogram one angle is 90 degrees then all angles are 90 degrees.

Consider a parallelogram &nbsp;ABCD, &nbsp;where it is given that &nbsp;LBCD = 90�.

{{{drawing( 160, 144, -10, 10, -10, 8, 
      line( -6,   -6,  4, -6), 
      line( -6,   -6, -2,  2), 
      line( -2,    2,  8,  2), 
      line(  8,    2,  4, -6),
      line( -6,   -6,  8,  2),
      line( -2,    2,  4, -6),
    locate( -6.5, -6.5, A),
    locate(  4.5, -6.5, B),
    locate(  8.5,  3.5, C),
    locate( -2.5,  4,   D),
    locate( -0,   -2.4, O)
)}}}

Since opposite angles in a parallelogram are equal, we have

&nbsp;LDAB = &nbsp;LBCD = 90�. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

Now, &nbsp;AB || DC.

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

&nbsp;LDAB + &nbsp;LADC=180�,

90� + &nbsp;LADC = 180�,

&nbsp;LADC = 180�-90� = 90�. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)

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

&nbsp;LABC = &nbsp;LADC = 90�. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3)

From equations &nbsp;(1), &nbsp;(2)&nbsp; and &nbsp;(3)

&nbsp;LBCD = &nbsp;LDAB = &nbsp;LADC = &nbsp;LABC = 90�.

Hence, &nbsp;all &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/Angle.wikipedia>angles</A>&nbsp; in a rectangle are equal to 90 degrees.

Proof 2. &nbsp;Diagonals of a rectangle are equal.

Consider two &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/Triangles-and-its-basic-properties.lesson>Triangles</A>&nbsp; &nbsp;ABD&nbsp; &nbsp;and&nbsp; &nbsp;ADC&nbsp; containing two diagonals &nbsp;BD&nbsp; and &nbsp;AC&nbsp; respectively.

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

From Pythagorean theorem,

In Triangle &nbsp;ABD,

{{{BD^2=(AB^2+AD^2)}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(4)

In Triangle &nbsp;ADC,

{{{AC^2=AD^2+DC^2}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(5)

From equations &nbsp;(5)&nbsp; and &nbsp;(6)

{{{BD^2=AC^2=(AD^2+DC^2)=(AB^2+AD^2)}}},

{{{BD^2=AC^2}}},

{{{BD=AC}}}.

Hence, &nbsp;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 &nbsp;diagonals of a rectangle are of equal length and bisect each other.

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

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

Triangles &nbsp;ABD&nbsp; and &nbsp;ADC&nbsp; are &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/Right-Triangles.lesson>Right Angled Triangles</A>&nbsp; right angled at &nbsp;LDAC&nbsp; and &nbsp;LADC&nbsp; respectively.

From &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Pythagorean-theorem/proof-of-pythagorean-theorem.lesson>Pythagorean theorem</A>,

In Triangle &nbsp;ABD,

In Triangle &nbsp;ADC,

From equations &nbsp;(7)&nbsp; and &nbsp;&nbsp;(8)

{{{BD^2=AC^2=(AD^2+DC^2)=(AB^2+AD^2)}}},

{{{BD^2=AC^2}}}.

QED