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© 2016. M. P. Chaudhary & Getachew Abiye Salilew. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Note on the Properties of Trapezoid
By M. P. Chaudhary & Getachew Abiye Salilew Madda Walabu University
Abstract- Gedefa and Chaudhary [2] introduce a new technique for identification of the nature of triangle. In this paper, we extended and applied this technique for the trapezoid.
Keywords: trapezoid, triangle.
GJSFR-F Classification: MSC 2010: 51A15, 51M20, 51M30.
NoteonthePropertiesofTrapezoid
Strictly as per the compliance and regulations of :
Global Journal of Science Frontier Research: FMathematics and Decision Sciences Volume 16 Issue 6 Version 1.0 Year 2016 Type : Double Blind Peer Reviewed International Research JournalPublisher: Global Journals Inc. (USA)Online ISSN: 2249-4626 & Print ISSN: 0975-5896
Note on the Properties of Trapezoid
M. P. Chaudhary α & Getachew Abiye Salilew σ
Author
α:
International Scientific Research and Welfare Organization, New Delhi 110018, India. e-mail: [email protected]
Author
σ:
Department of Mathematics, College of Natural and Computational Science, MaddaWalabu University, Bale Robe, Ethiopia.
Abstract-
Gedefa and Chaudhary [2] introduce a new technique for identification of the nature of
triangle. In this paper, we extended and applied this technique for the trapezoid.
Keywords:
trapezoid, triangle.
I.
Introduction
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Trapezoid is one of the interested areas in geometry, for the researcher since ancient time. But major contributions have been made on it during 19th centuries, which were initiated since 17th century. On this topic several contributions have been already done by the mathematician , but we believe that our approach for dealing current problems described in this article is different than others. Triangle is the simplest polygon with three edges and three vertices. It is one of the basic shapes in geometry. In Euclidean geometry any three points, when non- collinear, determine a unique triangle and a unique plane. The basic elements of any triangle are its sides and angles. Triangles are classified depending on relative sizes of their elements [1, 3].In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid in American and Canadian English but as a trapezium in English outside North America. A trapezium in Proclus’ sense is a quadrilateral having one pair of its opposite sides parallel.
In this study we consider a trapezoid which has only one pair of parallel sides. The parallel sides are called the bases, while the other sides are called the legs or lateral sides. The larger base side of a trapezoid used as simply the base of a trapezoid. When the legs have the same length and the base angles have the same measure then the trapezoid is acute angle trapezoid. If the two adjacent angles are right angle, then the trapezoid is a right angle trapezoid. If the trapezoid has no sides of equal measures, it is called a scalene trapezoid. In this paper, we presented acute angle trapezoid, right angle trapezoid and obtuse angle trapezoid which obtained from acute angle triangle, right angle triangle and obtuse angle triangle respectively.
Find necessary conditions, which enable to identify the nature of a trapezoid.
Our approach to derive necessary conditions, for identification the nature of a trapezoid, motivated by the recent work [2], and also by using known results.
II. Main Result
Theorem 1:
Proof:
[4- 13]
Ref
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edef
aNeg
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Fey
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. P
. C
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On I
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by N
ew A
ppro
ach,
Inte
rnat
ional
Res
earc
h J
ourn
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f P
ure
Alg
ebra
-5(9
),
2015
, 13
8-14
0.
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A B
CD
E
Note on the Properties of Trapezoid
Figure 1
Lemma 1:
Proof.
From the above figure-1, letAB = b = b + b , ≠Let us assume is parallel to and assume that ′ ′ is a longest side in the ∆ABE. Let ′ ′ be the length of the unique perpendicular segment from Eto . From, ∆ABE, we have the following conditions.
= = + ; = = + , ≠ ; = = + ; = = + ; = + ,
and clearly > .
≥ , ≥ ; = cos ; = sin ; = = + ; = − .
Using the work done in [2] we have the following.
= + − 2 cos
(1)
∆ABE is similar to ∆DCE. That is, ∆ABE~∆DCE.
Since AB ∥ DC, then ÐEDC = α, ÐDCE = γ.Thus, ∆ABE~∆DCE, by angle - angle - angle similarity theorem.
Therefore, by definition of similarity of triangles we have: = =Equivalently, we have
= = (2)
From this relation we get
= and = ⟹ = and == and = ⟹ + = and + =
⟹ c = and a = (3)
Similarly as equation (1), we have the following result from ∆DCE.
= + − 2 cos . (4)
Using (3) in to (4) we have
Notes
2.G
edefaN
egassaFey
issa an
d M
. P. C
hau
dhary
, On Id
entification
of th
e Natu
re of T
riangle b
y N
ew A
pproach
, I ntern
ational
Research
Journ
al of P
ure A
lgebra
-5(9), 2015, 138
-140.
Note on the Properties of Trapezoid
Figure 2
a = c + d − 2dc cos α ⟹ dab − d = dcb − d + d − 2d dcb − d cos α⟹ ( − ) = ( − ) + 1 − 2− cos⟹ = + ( − ) − 2 ( − ) cos .
(5)
Which is the general equation of any trapezoid.For = 90 we get from equations (4) and (5), = + and = + ( − ) respectively.Thus using [2], we have the following conditions from ∆DCE.
[A]. If = + then = 90 .[B]. If < + then is an acute angle.[C]. If > + then is an obtuse angle.
Equivalently we have three conditions for the trapezoid
[D]. If = + ( − ) then = 90 .[E]. If < + ( − ) then is an acute angle.[F]. If > + ( − ) then is an obtuse angle.
Let us give the name of a trapezoid based on the larger angle that lies on the larger base side from the two parallel sides of a trapezoid. Here and are angles that lie on the larger base side of a trapezoid . Angles and are also called base angles for a trapezoid and ≥ . So, based on the larger measure of angle from larger base side of a trapezoid we have the following.
[I]. For = 90 , the trapezoid is called right angle trapezoid.
AB
CD
E
= + ( − ) ; 0 < < = 90A
B
CD
−
[II]. For 0 < < 90 , the trapezoid is called an acute angle trapezoid.
< + ( − ) ;0 < ≤ < 90
E
A B
CD
AB
CD
Figure 3
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Ref
2.G
edef
aNeg
assa
Fey
issa
an
d M
. P
. C
hau
dhar
y,
On I
den
tifica
tion
of
the
Nat
ure
of
Trian
gle
by N
ew A
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ach,
Inte
rnat
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earc
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ure
Alg
ebra
-5(9
),
2015
, 13
8-14
0.
Note on the Properties of Trapezoid
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Corollary 1:Proof.
Corollary 2:
Proof.
Figure 4
[III]. For 90 < < 180 , the trapezoid is called an obtuse angle trapezoid.
E
AB
CD
> + ( − ) ; 0 < < 900; 900 < < 1800A
B
CD
Now we have the following observations from figure 2 to 4.
For a right angle trapezoid ABCD the acute angle = 45 ⟺ = − .Consider the right angle trapezoid ABCD in figure 2.
tan = ⟺ 1 = ⟺ = − ; ÐBCD = 135Similarly = 30 ⟺ = √ ( − )and = 60 ⟺ = √3( − ).
For a right angle trapezoid, ABCD, BD = √2bd ⟺ = .
Consider the right angle trapezoid ABCD in figure 2.
= + ( − ) = + − 2 + , because is a right angle trapezoid.
Let BD = √2bd then we get + = (BD) = 2 .
⟹ = + − 2 + ; ⟹ = 2 − 2 + , because + = (BD) .
⟹ = ⟺ = ; because all sides must be positive.Let =⇒ = + − 2 + ; ⇒ 0 = + − 2 ; ⇒ + = 2 = (BD) ; ⇒ BD = √2bdSimilarly, AC = √2bd ⟺ = ; = + 2 ( − ) ⟺ =If the two legs (lateral sides) of the trapezoid are equal then the trapezoid is called isosceles trapezoid.
An acute angle trapezoid ABCD is an isosceles trapezoid ⟺ 0 < = < 90 .
Consider an acute angle trapezoid ABCD in figure 3. Let = (trapezoid ABCD is an isosceles trapezoid). ⇒ sin = andsin = ; ⇒ ℎ = sin andℎ = sin ; ⇒ ℎ = sin = sin⟹ sin = sin , because = ; ⟹ =Therefore, 0 < = < 90 .Let, = . ⟹ sin = sin ; ⟹ = .Similarly when 0 < < < 90 an acute angle trapezoid ABCD, is not isosceles trapezoid.
An isosceles trapezoid ABCD is equilateral trapezoid (i.e. = = ), whenever =(1 + 2 cos ).
Corollary 3:
Proof.
Corollary 4:
Notes
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Let the trapezoid ABCD in figure 3 is an isosceles trapezoid.
⟹ =From equation (5), we get= + ( − ) − 2 ( − ) cos .⟹ 0 = ( − ) − 2 ( − ) cos , because =⟹ − = 2 cos ; ⟹ + 2 cos − = 2 cos ; ⟹ = .
(i). If, = + ( − ) , then trapezoid ABCD is right angle trapezoid.(ii). If, < + ( − ) , then trapezoid ABCD is an acute angle trapezoid.(iii). If, > + ( − ) , then trapezoid ABCD is an obtuse angle trapezoid.
Proof.
III. Conclusion
We have found the following conditions from our main result.
IV. Acknowledgement
The second named author is thankful to Dr. Habtamu Teka, president of Madda Walabu University, for his motivation towards research and providing academic environment within the university campus.
References Références Referencias
1. https://en.m.wikipedia.org/wiki/Triangle.2. GedefaNegassa Feyissa and M. P. Chaudhary, On Identification of the Nature of
Triangle by New Approach, International Research Journal of Pure Algebra-5(9), 2015, 138-140.
3. https://en.m.wikipedia.org/wiki/Trapezoid.4. M. P. Chaudhary, Lecture on Contribution of Indian Scholars in Mathematics,
Science and Philosophy at Department of Mathematics, Franklin & Marshal college, Lancaster, USA, 2007.
5. M. P. Chaudhary, Development of Mathematics from Sanskrit, Indian’s Intellectual Traditions and contribution to the world (Edited), D.K. Print World (P) Ltd.; New Delhi, 2010, 1-25.
6. P. Yiu, Isosceles triangles equal in perimeter and area, Missouri J. Math. Sci. 10 (1998), 106 - 111.
7. R. Homberger, Episodes of 19th and 20th century Euclidean Geometry, Math. Assoc. America, 1995.
8. C. Kimberling, Central points and central lines in the plane of a triangle, Math.
Magazine, 67 (1994), 163 – 187.9. P. Yiu, Introduction to the Geometry of the Triangle, Florida Atlantic University
lecturer notes, 2001.
10. M. S. Longuet – Higgins, Reflections on a triangle, part 2, Math. Gazette, 57 (1973)
293 – 296.
11. C. Kimberling, Major centers of triangles, Amer. Math. Monthly, 104 (1997) 431 –438.
12. C. Kimberling, Triangle centers and central triangle, Congressus Numerantium, 129
(1998). 1 – 285.
13. B. Scimemi, Paper – folding and Euler’s theorem revisited, Forum Geo, 2 (2002) 93
– 104.
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