Can Trapezoids Have Right Angles

Convex quadrilateral with at to the lowest degree i pair of parallel sides

Trapezoid (AmE) Trapezium (BrE)
Trapezoid or trapezium
Type	quadrilateral
Edges and vertices	four
Area	${\displaystyle {\tfrac {a+b}{2}}h}$
Properties	convex

Expect upward trapezoid in Wiktionary, the free dictionary.

A quadrilateral with at least i pair of parallel sides is called, in American and Canadian English, a trapezoid (). In British and other forms of English, a (Due north American) trapezoid is called a trapezium ().^{[1] [2]} The transposition of these two terms was a upshot of an fault in Charles Hutton'south mathematical lexicon.

A trapezoid is necessarily a convex quadrilateral in Euclidean geometry. The parallel sides are called the bases of the trapezoid. The other two sides are called the legs (or the lateral sides) if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases). A scalene trapezoid is a trapezoid with no sides of equal measure,^[3] in contrast with the special cases below.

Etymology and trapezium vs trapezoid [edit]

Hutton'southward mistake in 1795^[4]

Ancient Greek mathematician Euclid defined five types of quadrilateral, of which 4 had two sets of parallel sides (known in English every bit square, rectangle, rhombus and rhomboid) and the last did not accept two sets of parallel sides – a τραπέζια (trapezia ^[5] literally "a table", itself from τετράς (tetrás), "four" + πέζα (péza), "a foot; end, border, border").^[six]

Two types of trapezia were introduced by Proclus (412 to 485 AD) in his commentary on the first book of Euclid's Elements:^{[four] [7]}

• i pair of parallel sides – a trapezium (τραπέζιον), divided into isosceles (equal legs) and scalene (diff) trapezia
• no parallel sides – trapezoid (τραπεζοειδή, trapezoeidé, literally trapezium-like (εἶδος means "resembles"), in the same way equally cuboid means cube-like and rhomboid means rhombus-like)

All European languages follow Proclus'due south structure^{[7] [8]} equally did English language until the late 18th century, until an influential mathematical dictionary published by Charles Hutton in 1795 supported without explanation a transposition of the terms. This mistake was corrected in British English in nearly 1875, simply was retained in American English into the modern 24-hour interval.^[4]

Blazon	Sets of parallel sides	Image	Original terminology			Modern terminology
			Euclid (Definition 22)	Proclus (Definitions 30-34, quoting Posidonius)	Euclid / Proclus definition	British English (and European languages)	American English
Parallelogram	2		ῥόμβος (rhombos)		equilateral only not right-angled	Rhombus		Trapezoid (inclusive)
			ῥομβοειδὲς (rhomboides)		opposite sides and angles equal to one some other but not equilateral nor right-angled	Rhomboid (colloquially Parallelogram)
Non-parallelogram	ane		τραπέζια (trapezia)	τραπέζιον ἰσοσκελὲς (trapezion isoskelés)	2 parallel sides, and a line of symmetry	Isoceles Trapezium	Isoceles Trapezoid
				τραπέζιον σκαληνὸν (trapezion skalinón)	Two parallel sides, and no line of symmetry	Trapezium	Trapezoid (exclusive)
	0			τραπέζοειδὲς (trapezoides )	No parallel sides	Trapezoid	Trapezium

The shape is often called an irregular quadrilateral.^{[9] [10]}

Inclusive vs exclusive definition [edit]

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some ascertain a trapezoid equally a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.^[11] Others^[12] ascertain a trapezoid as a quadrilateral with at least i pair of parallel sides (the inclusive definition^[thirteen]), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is besides advocated in the taxonomy of quadrilaterals.

Under the inclusive definition, all parallelograms (including rhombuses, squares and non-square rectangles) are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses take mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices.

Special cases [edit]

Trapezoid special cases. The orange figures also qualify every bit parallelograms.

A right trapezoid (also called right-angled trapezoid) has 2 side by side correct angles.^[12] Correct trapezoids are used in the trapezoidal dominion for estimating areas nether a curve.

An astute trapezoid has ii adjacent acute angles on its longer base of operations border, while an obtuse trapezoid has i acute and i obtuse angle on each base.

An isosceles trapezoid is a trapezoid where the base angles take the same measure. Equally a consequence the two legs are also of equal length and it has reflection symmetry. This is possible for astute trapezoids or right trapezoids (rectangles).

A parallelogram is a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold rotational symmetry (or point reflection symmetry). It is possible for obtuse trapezoids or correct trapezoids (rectangles).

A tangential trapezoid is a trapezoid that has an incircle.

A Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two next right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the hyperbolic aeroplane has iii right angles.

Condition of existence [edit]

Four lengths a, c, b, d can plant the consecutive sides of a non-parallelogram trapezoid with a and b parallel only when^[fourteen]

${\displaystyle \displaystyle |d-c|<|b-a|<d+c.}$

The quadrilateral is a parallelogram when ${\displaystyle d-c=b-a=0}$ , merely it is an ex-tangential quadrilateral (which is not a trapezoid) when ${\displaystyle |d-c|=|b-a|\neq 0}$ .^{[15] : p. 35}

Characterizations [edit]

trapezoid/trapezium with opposing triangles ${\displaystyle S,\,T}$ formed by the diagonals

Given a convex quadrilateral, the post-obit backdrop are equivalent, and each implies that the quadrilateral is a trapezoid:

• It has two adjacent angles that are supplementary, that is, they add up to 180 degrees.
• The angle between a side and a diagonal is equal to the angle betwixt the opposite side and the aforementioned diagonal.
• The diagonals cut each other in mutually the same ratio (this ratio is the aforementioned equally that between the lengths of the parallel sides).
• The diagonals cut the quadrilateral into four triangles of which i opposite pair have equal areas.^{[15] : Prop.v}
• The product of the areas of the ii triangles formed by one diagonal equals the production of the areas of the two triangles formed past the other diagonal.^{[15] : Thm.6}
• The areas Southward and T of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation

${\displaystyle {\sqrt {G}}={\sqrt {Due south}}+{\sqrt {T}},}$

where K is the area of the quadrilateral.^{[15] : Thm.8}

• The midpoints of two opposite sides and the intersection of the diagonals are collinear.^{[15] : Thm.15}
• The angles in the quadrilateral ABCD satisfy ${\displaystyle \sin A\sin C=\sin B\sin D.}$ ^{[fifteen] : p. 25}
• The cosines of two next angles sum to 0, as do the cosines of the other two angles.^{[15] : p. 25}
• The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.^{[fifteen] : p. 26}
• 1 bimedian divides the quadrilateral into 2 quadrilaterals of equal areas.^{[15] : p. 26}
• Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.^{[15] : p. 31}

Additionally, the following backdrop are equivalent, and each implies that opposite sides a and b are parallel:

• The sequent sides a, c, b, d and the diagonals p, q satisfy the equation^{[15] : Cor.11}

${\displaystyle p^{2}+q^{2}=c^{ii}+d^{2}+2ab.}$

• The distance v between the midpoints of the diagonals satisfies the equation^{[15] : Thm.12}

${\displaystyle five={\frac {|a-b|}{ii}}.}$

Midsegment and acme [edit]

The midsegment (also called the median or midline) of a trapezoid is the segment that joins the midpoints of the legs. Information technology is parallel to the bases. Its length m is equal to the boilerplate of the lengths of the bases a and b of the trapezoid,^[12]

${\displaystyle thou={\frac {a+b}{2}}.}$

The midsegment of a trapezoid is one of the two bimedians (the other bimedian divides the trapezoid into equal areas).

The height (or altitude) is the perpendicular distance between the bases. In the case that the two bases have different lengths (a ≠ b), the height of a trapezoid h can exist determined past the length of its four sides using the formula^[12]

${\displaystyle h={\frac {\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2|b-a|}}}$

where c and d are the lengths of the legs.

Area [edit]

The expanse K of a trapezoid is given past^[12]

${\displaystyle K={\frac {a+b}{2}}\cdot h=mh}$

where a and b are the lengths of the parallel sides, h is the height (the perpendicular distance between these sides), and grand is the arithmetic mean of the lengths of the two parallel sides. In 499 AD Aryabhata, a peachy mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.8). This yields every bit a special case the well-known formula for the expanse of a triangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

The seventh-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with sequent sides a, c, b, d:

${\displaystyle 1000={\frac {ane}{2}}(a+b){\sqrt {c^{two}-{\frac {1}{iv}}\left((b-a)+{\frac {c^{2}-d^{2}}{b-a}}\right)^{ii}}}}$

where a and b are parallel and b > a.^[xvi] This formula tin be factored into a more symmetric version^[12]

${\displaystyle K={\frac {a+b}{4|b-a|}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}.}$

When one of the parallel sides has shrunk to a point (say a = 0), this formula reduces to Heron'due south formula for the expanse of a triangle.

Another equivalent formula for the area, which more closely resembles Heron's formula, is^[12]

${\displaystyle Thou={\frac {a+b}{|b-a|}}{\sqrt {(s-b)(s-a)(southward-b-c)(s-b-d)}},}$

where ${\displaystyle s={\tfrac {1}{two}}(a+b+c+d)}$ is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, only it differs from it, in that a trapezoid might not exist circadian (inscribed in a circumvolve). The formula is also a special instance of Bretschneider'south formula for a full general quadrilateral).

From Bretschneider's formula, it follows that

${\displaystyle Thou={\sqrt {{\frac {(ab^{2}-a^{ii}b-advertizement^{2}+bc^{2})(ab^{2}-a^{two}b-air conditioning^{2}+bd^{ii})}{four(b-a)^{ii}}}-\left({\frac {c^{2}+d^{ii}-a^{2}-b^{ii}}{four}}\correct)^{2}}}.}$

The line that joins the midpoints of the parallel sides, bisects the area.

Diagonals [edit]

The lengths of the diagonals are^[12]

${\displaystyle p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{ii}+bd^{2}}{b-a}}},}$

${\displaystyle q={\sqrt {\frac {ab^{2}-a^{2}b-advertizement^{2}+bc^{2}}{b-a}}}}$

where a is the short base of operations, b is the long base, and c and d are the trapezoid legs.

If the trapezoid is divided into four triangles by its diagonals Air conditioning and BD (as shown on the correct), intersecting at O, and then the expanse of ${\displaystyle \triangle }$ AOD is equal to that of ${\displaystyle \triangle }$ BOC , and the product of the areas of ${\displaystyle \triangle }$ AOD and ${\displaystyle \triangle }$ BOC is equal to that of ${\displaystyle \triangle }$ AOB and ${\displaystyle \triangle }$ COD . The ratio of the areas of each pair of adjacent triangles is the aforementioned as that between the lengths of the parallel sides.^[12]

Allow the trapezoid take vertices A, B, C, and D in sequence and have parallel sides AB and DC. Allow E be the intersection of the diagonals, and let F exist on side DA and Yard be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic hateful of AB and DC:^[17]

${\displaystyle {\frac {1}{FG}}={\frac {1}{two}}\left({\frac {1}{AB}}+{\frac {one}{DC}}\right).}$

The line that goes through both the intersection signal of the extended nonparallel sides and the intersection bespeak of the diagonals, bisects each base of operations.^[18]

Other properties [edit]

The heart of expanse (center of mass for a compatible lamina) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular altitude x from the longer side b given by^[19]

${\displaystyle x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).}$

The eye of area divides this segment in the ratio (when taken from the short to the long side)^{[20] : p. 862}

${\displaystyle {\frac {a+2b}{2a+b}}.}$

If the angle bisectors to angles A and B intersect at P, and the angle bisectors to angles C and D intersect at Q, so^[xviii]

${\displaystyle PQ={\frac {|Advertizement+BC-AB-CD|}{2}}.}$

Applications [edit]

Compages [edit]

In architecture the discussion is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in Egyptian style. If these accept straight sides and sharp angular corners, their shapes are unremarkably isosceles trapezoids. This was the standard style for the doors and windows of the Inca.^[21]

Geometry [edit]

The crossed ladders problem is the trouble of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the altitude from the perpendicular leg to the diagonal intersection.

Biology [edit]

In morphology, taxonomy and other descriptive disciplines in which a term for such shapes is necessary, terms such as trapezoidal or trapeziform commonly are useful in descriptions of particular organs or forms.^[22]

Computer engineering [edit]

In computer engineering, specifically digital logic and calculator architecture, trapezoids are typically utilized to symbolize multiplexors. Multiplexors are logic elements that select between multiple elements and produce a single output based on a select signal. Typical designs volition employ trapezoids without specifically stating they are multiplexors equally they are universally equivalent.

See also [edit]

• Frustum, a solid having trapezoidal faces
• Polite number, also known as a trapezoidal number
• Wedge, a polyhedron defined by 2 triangles and three trapezoid faces.

References [edit]

1. ^ http://www.mathopenref.com/trapezoid.html Mathopenref definition
2. ^ A. D. Gardiner & C. J. Bradley, Plane Euclidean Geometry: Theory and Bug, UKMT, 2005, p. 34.
3. ^ Types of quadrilaterals
4. ^ ^{a b c} James A. H. Murray (1926). A New English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society. Vol. X. Clarendon Press at Oxford. p. 286 (Trapezium). With Euclid (c 300 B.C.) τραπέζιον included all quadrilateral figures except the square, rectangle, rhombus, and rhomboid; into the varieties of trapezia he did not enter. Merely Proclus, who wrote Commentaries on the Start Volume of Euclid's Elements A.D. 450, retained the name τραπέζιον only for quadrilaterals having 2 sides parallel, subdividing these into the τραπέζιον ἰσοσκελὲς, isosceles trapezium, having the 2 non-parallel sides (and the angles at their bases) equal, and σκαληνὸν τραπέζιον, scalene trapezium, in which these sides and angles are unequal. For quadrilaterals having no sides parallel, Proclus introduced the name τραπέζοειδὲς TRAPEZOID. This nomenclature is retained in all the continental languages, and was universal in England till late in the 18th century, when the application of the terms was transposed, so that the figure which Proclus and modern geometers of other nations phone call specifically a trapezium (F. trapèze, Ger. trapez, Du. trapezium, It. trapezio) became with most English writers a trapezoid, and the trapezoid of Proclus and other nations a trapezium. This changed sense of trapezoid is given in Hutton'southward Mathematical Dictionary, 1795, as 'sometimes' used -- he does not say by whom; only he himself unfortunately adopted and used it, and his Dictionary was doubtless the chief agent in its improvidence. Some geometers however continued to employ the terms in their original senses, and since c 1875 this is the prevalent use.
5. ^ Euclid Elements Book I Definition 22
6. ^ πέζα is said to be the Doric and Arcadic form of πούς "foot", but recorded merely in the sense "instep [of a man human foot]", whence the significant "border, edge". τράπεζα "table" is Homeric. Henry George Liddell, Robert Scott, Henry Stuart Jones, A Greek-English language Lexicon, Oxford, Clarendon Press (1940), s.5. πέζα, τράπεζα.
7. ^ ^{a b} Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (5 April 2016). The Symmetries of Things. CRC Printing. p. 286. ISBN978-1-4398-6489-0.
8. ^ For example: French trapèze, Italian trapezio, Portuguese trapézio, Spanish trapecio, German language Trapez, Ukrainian "трапеція", e.g. "Larousse definition for trapézoïde".
9. ^ Chambers 21st Century Dictionary Trapezoid
10. ^ "1913 American definition of trapezium". Merriam-Webster Online Dictionary . Retrieved 2007-12-10 .
11. ^ "American School definition from "math.com"". Retrieved 2008-04-14 .
12. ^ ^{a b c d e f m h i} Weisstein, Eric W. "Trapezoid". MathWorld.
13. ^ Trapezoids, [ane]. Retrieved 2012-02-24.
14. ^ Ask Dr. Math (2008), "Area of Trapezoid Given Only the Side Lengths".
15. ^ ^{a b c d due east f g h i j thou 50} Martin Josefsson, "Characterizations of trapezoids", Forum Geometricorum, xiii (2013) 23-35.
16. ^ T. K. Puttaswamy, Mathematical achievements of pre-modern Indian mathematicians, Elsevier, 2012, p. 156.
17. ^ GoGeometry, [2]. Retrieved 2012-07-08.
18. ^ ^{a b} Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, p. 55.
19. ^ efunda, General Trapezoid, [iii]. Retrieved 2012-07-09.
20. ^ Tom M. Apostol and Mamikon A. Mnatsakanian (December 2004). "Figures Circumscribing Circles" (PDF). American Mathematical Monthly. 111 (10): 853–863. doi:x.2307/4145094. JSTOR 4145094. Retrieved 2016-04-06 .
21. ^ "Machu Picchu Lost City of the Incas - Inca Geometry". gogeometry.com . Retrieved 2018-02-13 .
22. ^ John 50. Capinera (xi August 2008). Encyclopedia of Entomology. Springer Science & Business Media. pp. 386, 1062, 1247. ISBN978-1-4020-6242-1.

Farther reading [edit]

• D. Fraivert, A. Sigler and Chiliad. Stupel : Mutual properties of trapezoids and convex quadrilaterals

External links [edit]

• "Trapezium" at Encyclopedia of Mathematics
• Weisstein, Eric W. "Right trapezoid". MathWorld.
• Trapezoid definition   Area of a trapezoid   Median of a trapezoid With interactive animations
• Trapezoid (North America) at elsy.at: Animated form (construction, circumference, surface area)
• Trapezoidal Rule on Numerical Methods for Stalk Undergraduate
• Autar Kaw and E. Eric Kalu, Numerical Methods with Applications, (2008)

Can Trapezoids Have Right Angles,

Source: https://en.wikipedia.org/wiki/Trapezoid

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