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Trapezoids Talk- A paper by Michael Keyton
by Total Gadha - Wednesday, 24 September 2008, 07:25 PM
 

cat cat 2008 cat 2009 trapeziumWhile I am busy writing the next article for TG.com, here is an offering by another mathematician, Michael Keyton of Illinois Mathematics and Science Academy. In this paper, Mr. Keyton discusses about the properties of a trapezium. Some of the properties are known whereas many of these are unknown. I discovered the paper while searching for one of the properties of the trapezium, namely the sum of the squares of the diagonals of a trapezium. Mr. Keyton has graciously given us the permission to publish the paper on TG. So here is it everyone, the complete compendium of the properties of a trapezium. Include them in your armory of formula.

 

Trapezoids Talk

In every high school textbook, the trapezoid in included as one of the quadrilaterals to study (or investigate). In almost every one, it is defined as a quadrilateral with exactly one pair of parallel sides. In this talk, I will argue that the definition should be changed and that there is much more to the trapezoid than is given in the books.

Almost every theorem about a trapezoid can be broken into two categories – those that are really about parallel sides and those that in some way incorporate a feature of a quadrilateral.

In the history of geometry, many definitions have been changed, some from being exclusive to being inclusive. An exclusive definition is one that separates other objects from being in the same class. For example, in Euclid, an isosceles triangle is defined to be a triangle with exactly two equal sides. Thus, an equilateral triangle is not isosceles. Likewise all his definitions of quadrilaterals are exclusive. During the next 2300 years, most of these have been changed. Now in every high school textbook, equilateral triangles are isosceles, rectangles and rhombi are parallelograms, and squares are rectangles and rhombi. There are two advantages to having inclusive definitions – (1) theorems for the more restricted case become corollaries for the more general case and (2) converses do not need to contain an “or” conclusion.

By one construction, we can easily see that the parallelogram is a special case of a trapezoid. Take two points on each of two parallel lines. The quadrilateral formed by using these points is a trapezoid (or a parallelogram). Consequently, the definition can be strengthened by including it as such. So why is the definition maintained in the textbooks? I think primarily that many authors do not wish to go against standard terminology, most of the authors have not thought about the inconsistency in terminology, they are not actively engaged in discovering and proving theorems in geometry, and there are no converses for the trapezoid covered in high school geometry.

Thus, for good mathematical reasons let’s change the definition to:

A trapezoid is a quadrilateral with at least one pair of parallel sides.

In the theorems that follow, some require that a pair of sides be non-parallel, but the parallel case follows as well, usually with little or no additional proof.

While we are at this change, the same argument applies to the isosceles trapezoid and the rectangle. A simple construction shows that the rectangle is a special case of the isosceles trapezoid. How then can we define the isosceles trapezoid so that the rectangle is a special case. I offer a variety of different definitions.

(1) An isosceles trapezoid is a cyclic trapezoid.
(2) An isosceles trapezoid is a trapezoid with a pair of supplementary opposite angles.
(3) An isosceles trapezoid is a trapezoid with the other pair of sides anti-parallel with respect to the parallel sides.
(4) An isosceles trapezoid is a trapezoid with a pair of congruent base angles.
(5) An isosceles trapezoid is a trapezoid with congruent diagonals.

Here are a few other definitions that I offer.

The diacenter of a quadrilateral is the intersection point of the diagonals.
A quord of a quadrilateral is a segment with endpoints on two sides of a quadrilateral.
A median is a quord with endpoints the midpoints on opposite sides.

cat cat 2008 cat 2009 trapezium

cat cat 2008 cat 2009 trapezium

Re: Trapezoids Talk- A paper by Michael Keyton
by Gul Gul - Wednesday, 24 September 2008, 11:28 PM
  Long time.......smile
Re: Trapezoids Talk- A paper by Michael Keyton
by Total Gadha - Wednesday, 24 September 2008, 11:38 PM
  True...but we won't upload an article on TG till it's useful. TG won't become a news channel. smile
Re: Trapezoids Talk- A paper by Michael Keyton
by vamsi krishna - Thursday, 25 September 2008, 10:37 AM
  aaaaaaaaaaaaaaaaaaaaaaaaaahhhhhhhhhh

Conquering Trepezoid

Thanks TG sir

VaMsI
Re: Trapezoids Talk- A paper by Michael Keyton
by Krushang Shah - Thursday, 25 September 2008, 11:27 AM
  Thanks A Lot TG!!!

Excellent as Usual...

Perfection Exists.....
Re: Trapezoids Talk- A paper by Michael Keyton
by avinanda dutta - Thursday, 25 September 2008, 01:51 PM
  where is the srticle for permutation. i was simply in love with it.
Re: Trapezoids Talk- A paper by Michael Keyton
by inderjeet singh - Thursday, 25 September 2008, 03:16 PM
 

very useful

thanks sir

Re: Trapezoids Talk- A paper by Michael Keyton
by Total Gadha - Thursday, 25 September 2008, 03:20 PM
  Hi Avinanda,

Find the article over here:

http://totalgadha.com/mod/forum/discuss.php?d=3537

Total Gadha
Re: Trapezoids Talk- A paper by Michael Keyton
by Arindam Coomar - Thursday, 25 September 2008, 07:24 PM
  Thanks a lot TG for the wonderful article... Really many things to learn in trapeziod too...
Re: Trapezoids Talk- A paper by Michael Keyton
by chandra vikas - Friday, 26 September 2008, 03:49 AM
  thanks TG for this helpful article on trapezoids....
helped a lot..but if only you could write on the study plan for these 50 days, it would be great bcoz the study plan covers a very long time schedule.
once again thanx for this article...
Re: Trapezoids Talk- A paper by Michael Keyton
by nitin . - Friday, 26 September 2008, 07:54 AM
  Tg Sir plz post a article on time n work.
Re: Trapezoids Talk- A paper by Michael Keyton
by Prashant Chanchal - Saturday, 27 September 2008, 02:15 PM
  Hi TG,
I guess there is some error in the 3rd property mentioned (High-school) where it says the Area = Half of (Height x Median), which should be Half of (Height x Median).
Thanks
Re: Trapezoids Talk- A paper by Michael Keyton
by Total Gadha - Monday, 29 September 2008, 12:40 AM
  Hi Prashant,

Corrected. smile

Total Gadha
Re: Trapezoids Talk- A paper by Michael Keyton
by Mou Sukoshi - Tuesday, 30 September 2008, 11:21 AM
 

Sir,

Thanks a lot for such a nice article.

But I have a few confusions.

1> From 15 b) What are harmonic sets?

2> On the net, I found 2 contradictory definitions for trapezoid and trapezium.

  British USA
Trapezoid A quadrilateral with no sides parallel A quadrilateral with one pair of parallel sides
Trapezium A quadrilateral with one pair of parallel sides A quadrilateral with no sides parallel

In school, we learnt the British defn. I am confused as to which defn is to be followed for CAT? Or can we interchangeably use trapezoid and trapezium to mean a quadrilateral with atleast one pair of parallel sides?


Re: Trapezoids Talk- A paper by Michael Keyton
by Abhimanyu Kumar - Saturday, 4 October 2008, 02:16 AM
  Hi TG,

An insightful article. thanks.

But I dont know I feel like there is a discrepancy in the (15)th theorem.

LB/LA = AB/CD .

This seems to me that it contradicts my intuition (it doesnt follow rule of symmetry)
What I mean by symmetry is that the rule doesnt identifies in itself that why LA/LB = AB/CD wont hold true why only LB/LA = AB/CD  is true.

The second one looks perfect (HA/HC = AB/CD) from that point of view.
Can you please clarify.

Also I feel that TG is  a little slow, takes comparatively more time to upload in a browser. So if possible do something about it. Its a feedback from my side because I love your website

-abhimanyu
Re: Trapezoids Talk- A paper by Michael Keyton
by DINKAR MISHRA - Tuesday, 14 October 2008, 06:38 PM
  good collection
thanks T.G
Re: Trapezoids Talk- A paper by Michael Keyton
by ashish kumar - Tuesday, 14 October 2008, 07:13 PM
 

Thank a lot..

 

Re: Trapezoids Talk- A paper by Michael Keyton
by santhana lakshmanan - Tuesday, 15 September 2009, 05:47 AM
  Hi TG,

In point(15) shouldn't the formula LB/LC be actually LB/LA..Correct me if I am wrong..
Re: Trapezoids Talk- A paper by Michael Keyton
by Subhash Medhi - Sunday, 20 June 2010, 11:43 PM
  Dear TG sir,
                I think previous formula given in point(15) was correct. It should be LB/LC = AB/CD and not LB/LA = AB/CD.

Triangles, LAB and LDC are similar. So, it follows naturally.Kindly correct me if i am wrong.

Regards,
Subhash
Re: Trapezoids Talk- A paper by Michael Keyton
by Amal Nicholas - Saturday, 27 November 2010, 12:16 AM
  Seriously, most of this is not required for CAT. I think "maxima and minima Inequalities-basics" and "Remainders", are two of the BEST articles EVER! They were useful in almost all the mocks I wrote and eventually, even in my actual CAT paper. Thank u so much TG!
Re: Trapezoids Talk- A paper by Michael Keyton
by neha aggarwal - Saturday, 23 June 2012, 06:51 PM
  Hi TG Sir smile,

Nice article smile

I am unable to find any article on basic geometry concepts as whole please provide the links.. Searched for them a lot..