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Area of Rectangle of size 30 x 7 and Geometric Properties like symmetry,perimeter of rectangle,diagonals






We are given that the length of the rectangle = 30.0 units and the width = 7.0 units. 
AD = BC = 30.0 units.
And AB = CD = 7.0 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) = 30.0 x 7.0 = 210.0 square units

Calculating the perimeter of the rectangle

Perimeter of this rectangle = 2 * (length + width ) = 2 * (30.0 + 7.0) = 74.0 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 =  (30.02 + 7.02)(1/2)
So the length of the diagonal = 30.81 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 = 30.81/2 units = 15.4 units
Area of the circum-circle = PI x circumradius2 = PI * 15.4 * 15.4 = 745.34 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 = 7.0/30.0 = 0.23
Angle CAD = Angle ACB = tan-1 0.23  = 0.23 radians = 13.13 degrees

And Angle BAC = Angle ACD = 90 degrees - Angle CAD   = 1.34 radians  = 76.87 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 = 7.0 units (opposite sides of a rectangle are equal) 
(d) AD = BC  = 30.0 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 = 7.0 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. 



To understand more about the geometric features and properties of rectangles, formulas related to mensuration etc. you might find it useful to read the properties of a Rectangle tutorial over here. Many of these concepts are a part of the Grade 9 and 10 Mathematics syllabus of the UK GCSE curriculum, Common Core Standards in the US, ICSE/CBSE/SSC syllabus in India. You may check out our free and printable worksheets for Common Core and GCSE.
 


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