How to get ,The Area of Trapezium from Triangle

Lot of mathematicians have proved Pythagoras theorem in their own ways. If you google it you will indeed found hundred of ways.

Meanwhile I was also sure that maybe one day I could find something new out of this incredible Pythagoras theorem and Recently I got something which I would like to share with you.

To Prove: Deriving the equation of area of trapezium using Arcs

Proof: There is a triangle ABC with sides a b and c as shown in the figure.

Now,

Area of ∆  BCEG = Area of ∆  BDC +Area of ⌂ DCEF + Area of ∆ EFG

c^2=ac/2+ Area of ⌂ DCEF + (c-b)  c/2

(2c^2– ac –c^2+ bc )/2=Area of ⌂ DCEF

(c^2– ac+ bc )/2=Area of ⌂ DCEF

c(c– a+ b)/2=Area of ⌂ DCEF

Area of ⌂ DCEF=BC(DE+CF)/2

Copyrighted©PiyushGoel

oh!         very nice to see............. there is no comments .................wonderful.

1 hour ago, mathspassion said:

oh!         very nice to see............. there is no comments .................wonderful.

What would you expect us to say?

Congratulations you have derived the conventioanl formula for the area of a trapezium as being the average length of the parallel sides times the perpendicular distance between them.

Incidentally I would check the first line of your proof for what I assume is a typographical error.
BCEG is not a triangle as stated. Also the letters are not in cyclic order round the square.

Edited May 11, 20187 yr by studiot

Congrats on the minor discovery. Any discovery however little, is significant.

 

But the next thing I would do is see where the discovery applies. It may seem like a simple polygon, but it is also a geometric construction. (What I am calling is using drawing and geometry to form shapes with measurable properties. Like dividing a line in half with 2 arcs of a compass.)

 

I know it seems like the use for drawing this way is minor. However, what if this drawing method was applied to computer graphics. Imagine drawing a polygon mesh in 3DS Max and have that mesh be measurable by your polygons.

 

It is difficult to come up with a major application, because this isn’t my design, but if you expand it to polygons of more sides with the inside of those polygons measured; you’d have a brick to build a polygon block and measure it at the same time. Now that isn’t a little discovery.

21 hours ago, studiot said:

What would you expect us to say?

Congratulations you have derived the conventioanl formula for the area of a trapezium as being the average length of the parallel sides times the perpendicular distance between them.

Incidentally I would check the first line of your proof for what I assume is a typographical error.
BCEG is not a triangle as stated. Also the letters are not in cyclic order round the square.

yes right BCGE is not triangle it is square and second letters not in cyclic order ....sir thnxs a lot for mentioning two typographical error........next time ....never .