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Area of Rectangle of size 8 x 6 and Geometric Properties like symmetry,perimeter of rectangle,diagonals






We are given that the length of the rectangle = 8.0 units and the width = 6.0 units. 
AD = BC = 8.0 units.
And AB = CD = 6.0 units. 

The units may be any units of length: inches, cm, feet, miles, km etc.

Geometric Properties we'll compute 

Let us compute the area, perimeter, length of the diagonal, circumradius, area of the circumcircle and the angle(s) made by the diagonal with the sides. 


Calculating the area of the rectangle
Area of this rectangle = length x width (of length x breadth) = 8.0 x 6.0 = 48.0 square units

Calculating the perimeter of the rectangle

Perimeter of this rectangle = 2 * (length + width ) = 2 * (8.0 + 6.0) = 28.0 units

Calculating the length of the diagonal of the rectangle

The diagonal of this rectangle may be computed using Pythagorean Theorem (or Pythagoras Theorem). 
Triangles ADC (or BDC) are both right angled triangles. 
So, the diagonal(s) may be seen as the hypotenuse of a right triangle. So the square of the length of the diagonal equals the sum of the squares of the length and the width. 
Length of the diagonals AC and BD =  (8.02 + 6.02)(1/2)
So the length of the diagonal = 10.0 units

Computing the circumradius and the area of the circumcircle

A rectangle doesn't really have an incircle which touches each side. That is possible only in the limiting case when the length and width are equal and it is a square. 
However, it is indeed possible to draw a circle passing through all 4 vertices of a rectangle. Both the diagonals are diameters of the circle. 
Angle in a semi-circle is a right angle: each of the 4 angles of the rectangle become angles of a semi-circle when we draw the circumcircle. 

Radius of the circum-circle = half the length of the diagonal = 10.0/2 units = 5.0 units
Area of the circum-circle = PI x circumradius2 = PI * 5.0 * 5.0 = 78.54 square units


Symmetry and Axes of Symmetry

A rectangle has 2 axes of symmetry: a line passing through the midpoints of AB and CD  and another line passing through the midpoint of AD and BC. 
Both pass through the point E. A rectangle is also "isogonal" in nature, though this is a concept you might study or encounter later. 


Calculating the angle made by the diagonal with the sides

We can use a little bit of trigonometry to calculate this. 
Tan of Angle CAD = Tan of Angle ACB  = width of rectangle / length of rectangle = 6.0/8.0 = 0.75
Angle CAD = Angle ACB = tan-1 0.75  = 0.64 radians = 36.87 degrees

And Angle BAC = Angle ACD = 90 degrees - Angle CAD   = 0.93 radians  = 53.13 degrees

Examples of congruence and congruent triangles

Each diagonal divides the rectangle into a pair of congruent triangles. 

Diagonal BD divides the rectangle into Triangle BAD and Triangle BCD which are congruent. 
Proof:
(a) Angle BAD = Angle BCD = 90 degrees (Right angle)
(b) BD is common side to both (Hypotenuse)
(c) BA = CD = 6.0 units (opposite sides of a rectangle are equal) 
(d) AD = BC  = 8.0 units (opposite sides of a rectangle are equal) 

Using (a) (b) and (c) the two triangles are congruent using RHS or HL (hypotenuse-leg) congruence.
Or using (c) (a) and (d) the two triangles are congruent using SAS congruence (Side-angle-side)

Similarly, we can prove that AC divides the rectangle into two congruent triangles. 

Also, the two diagonals divide the rectangle into two pairs of congruent triangles
 (Triangle AEB and Triangle CED are congruent, Triangle BEC and Triangle AED are congruent)

To prove that Triangle AEB and Triangle CED are congruent:
(a) AB = CD = 6.0 units (Opposite sides of a rectangle are equal)
(b) Angle ABD = Angle CDB (AB is parallel to CD and BD is a transversal cutting them, these are alternate angles, so equal)
(c) Angle BAC = Angle DCA (similar reason as above)
(d) Angle BEA = Angle CED (opposite angles or vertical angles are equal)

So we can prove the two triangles congruent by ASA Congruence (using (c) (a) (b)) 
Or we can prove the two triangles congruent by AAS Congruence (using (d)(b)(a)) 

Similarly we can prove Triangles BEC and AED to be congruent. 

Geometric Properties of Rectangles


Remember: a rectangle is a quadrilateral with 4 right angles. Opposite sides are equal. All properties which apply to parallelograms apply to rectangles. 
A square may be studied as a special case of a rectangle where all sides are equal. 
Every rectangle is a convex polygon and a cyclic quadrilaterial where the diagonal is a diameter of the circumcircle. 



Example of a Pythagorean triple or triplet

Notice how the width (6), length (8), diagonal (10.0) lengths are all three integers. This is an example of Pythagorean triples. The square of the diagonal equals the sum of the squares of the length and the width. These are not, however, primitive pythagorean triples as they are multiples of (3,4,5).


Some example(s) to illustrate similar rectangles, size transformation, scale factor and geometrical similarity.


These tutorials on Similar Triangles, Scale Factor and Size Transformationmight also be useful.

Some example(s) to illustrate similar rectangles, size transformation, scale factor and geometrical similarity

Example:

Let the current rectangle (of dimensions 8.0 x 6.0) be rectangle A.

Both these rectangles are geometrically similar because the ratio of lengths of corresponding sides are equal. 
Length of rectangle A : Length of rectangle B = Width of rectangle A : Width of rectangle B
8.0 : 4.0 = 6.0 : 3.0 = 2.0 : 1

And the ratio of their areas:
Area of rectangle A : Area of rectangle B = 48.0 : 12.0  = 4.0 : 1

The scale factor between A:B  = Length of rectangle A : Length of rectangle B = 8.0 : 4.0 = 2.0

Note that the ratio of the areas is the square of the ratio of the corresponding sides (or the scale factor). 

This is an example of size transformation in 2D figures: 
when all sides of figure are multiplied by a ratio R the area of the new figure is R2 times the area of the original figure.



















Some more example(s):


Geometric Properties of a rectangle of size 9 x 6.



Geometric Properties of a rectangle of size 9 x 7.




 


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