Isosceles trapezoid

Isosceles trapezoid
Isosceles trapezoid
	Isosceles trapezoid with axis of symmetry
Type	quadrilateral, trapezoid
Edges and vertices	4
Symmetry group	Dih1, [ ], (*), order 1
Properties	convex, cyclic
Dual polygon	Kite

Last updated May 10, 2024

In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure,^[1] or as a trapezoid whose diagonals have equal length.^[2] Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (properties shared with the parallelogram), and the diagonals have equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base).

Special cases

Rectangles and squares are usually considered to be special cases of isosceles trapezoids though some sources would exclude them.^[3]

Another special case is a 3-equal side trapezoid, sometimes known as a trilateral trapezoid^[4] or a trisosceles trapezoid. They can also be seen dissected from regular polygons of 5 sides or more as a truncation of 4 sequential vertices.

Self-intersections

Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite.^[5] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms, crossed quadrilaterals in which opposite sides have equal length.

Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides (or either pair of opposite sides in the case of a rectangle) of an isosceles trapezoid.^[6]

Convex isosceles trapezoid	Crossed isosceles trapezoid	antiparallelogram

Characterizations

If a quadrilateral is known to be a trapezoid, it is not sufficient just to check that the legs have the same length in order to know that it is an isosceles trapezoid, since a rhombus is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry through the midpoints of opposite sides.

Any one of the following properties distinguishes an isosceles trapezoid from other trapezoids:

The diagonals have the same length.
The base angles have the same measure.
The segment that joins the midpoints of the parallel sides is perpendicular to them.
Opposite angles are supplementary, which in turn implies that isosceles trapezoids are cyclic quadrilaterals.
The diagonals divide each other into segments with lengths that are pairwise equal; in terms of the picture below, AE = DE, BE = CE (and AE ≠ CE if one wishes to exclude rectangles).

Angles

In an isosceles trapezoid, the base angles have the same measure pairwise. In the picture below, angles ∠ABC and ∠DCB are obtuse angles of the same measure, while angles ∠BAD and ∠CDA are acute angles, also of the same measure.

Since the lines AD and BC are parallel, angles adjacent to opposite bases are supplementary, that is, angles ∠ABC + ∠BAD = 180°.

Diagonals and height

Another isosceles trapezoid. Isoscelestriangle2.svg — Another isosceles trapezoid.

The diagonals of an isosceles trapezoid have the same length; that is, every isosceles trapezoid is an equidiagonal quadrilateral. Moreover, the diagonals divide each other in the same proportions. As pictured, the diagonals AC and BD have the same length (AC = BD) and divide each other into segments of the same length (AE = DE and BE = CE).

The ratio in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, that is,

{\frac {AE}{EC}}={\frac {DE}{EB}}={\frac {AD}{BC}}.

The length of each diagonal is, according to Ptolemy's theorem, given by

p={\sqrt {ab+c^{2}}}

where a and b are the lengths of the parallel sides AD and BC, and c is the length of each leg AB and CD.

The height is, according to the Pythagorean theorem, given by

h={\sqrt {p^{2}-\left({\frac {a+b}{2}}\right)^{2}}}={\tfrac {1}{2}}{\sqrt {4c^{2}-(a-b)^{2}}}.

The distance from point E to base AD is given by

d={\frac {ah}{a+b}}

where a and b are the lengths of the parallel sides AD and BC, and h is the height of the trapezoid.

Area

The area of an isosceles (or any) trapezoid is equal to the average of the lengths of the base and top (the parallel sides) times the height. In the adjacent diagram, if we write AD = a, and BC = b, and the height h is the length of a line segment between AD and BC that is perpendicular to them, then the area K is

K={\tfrac {1}{2}}\left(a+b\right)h.

If instead of the height of the trapezoid, the common length of the legs AB =CD = c is known, then the area can be computed using Brahmagupta's formula for the area of a cyclic quadrilateral, which with two sides equal simplifies to

K={\sqrt {(s-a)(s-b)(s-c)^{2}}},

where $s={\tfrac {1}{2}}(a+b+2c)$ is the semi-perimeter of the trapezoid. This formula is analogous to Heron's formula to compute the area of a triangle. The previous formula for area can also be written as

K={\tfrac {1}{4}}{\sqrt {(a+b)^{2}(a-b+2c)(b-a+2c)}}.

Circumradius

The radius in the circumscribed circle is given by^[7]

R=c{\sqrt {\frac {ab+c^{2}}{4c^{2}-(a-b)^{2}}}}.

In a rectangle where a = b this is simplified to $R={\tfrac {1}{2}}{\sqrt {a^{2}+c^{2}}}$ .

Related Research Articles

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices $,, and is sometimes denoted as .$

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.

<span class="mw-page-title-main">Right triangle</span> Triangle containing a 90-degree angle

A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle.

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.

<span class="mw-page-title-main">Kite (geometry)</span> Quadrilateral symmetric across a diagonal

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

<span class="mw-page-title-main">Parallelogram</span> Quadrilateral with two pairs of parallel sides

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

<span class="mw-page-title-main">Bisection</span> Division of something into two equal or congruent parts

In geometry, bisection is the division of something into two equal or congruent parts. Usually it involves a bisecting line, also called a bisector. The most often considered types of bisectors are the segment bisector, a line that passes through the midpoint of a given segment, and the angle bisector, a line that passes through the apex of an angle . In three-dimensional space, bisection is usually done by a bisecting plane, also called the bisector.

<span class="mw-page-title-main">Rhombus</span> Quadrilateral with sides of equal length

In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

<span class="mw-page-title-main">Trapezoid</span> Convex quadrilateral with at least one pair of parallel sides

In geometry, a trapezoid in North American English, or trapezium in British English, is a quadrilateral that has at least one pair of parallel sides.

<span class="mw-page-title-main">Isosceles triangle</span> Triangle with at least two sides congruent

In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four sides of equal length and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted $ABCD .$

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.

In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

In Euclidean geometry, Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon parallelogram. It is named after Pierre Varignon, whose proof was published posthumously in 1731.

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral.

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the legs. The legs can be equal, but they don't have to be.

In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.

In Euclidean geometry, a right kite is a kite that can be inscribed in a circle. That is, it is a kite with a circumcircle. Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals, since all kites have an incircle. One of the diagonals divides the right kite into two right triangles and is also a diameter of the circumcircle.

References

↑ "Trapezoid - math word definition - Math Open Reference".
↑ Ryoti, Don E. (1967). "What is an Isosceles Trapezoid?". The Mathematics Teacher. 60 (7): 729–730. doi:10.5951/MT.60.7.0729. JSTOR 27957671.
↑ Larson, Ron; Boswell, Laurie (2016). Big Ideas MATH, Geometry, Texas Edition. Big Ideas Learning, LLC (2016). p. 398. ISBN 978-1608408153.
↑ "A Hierarchical Classification of Quadrilaterals". dynamicmathematicslearning.com. Retrieved February 10, 2024.
↑ Halsted, George Bruce (1896), "Chapter XIV. Symmetrical Quadrilaterals", Elementary Synthetic Geometry, J. Wiley & sons, pp. 49–53.
↑ Whitney, William Dwight; Smith, Benjamin Eli (1911), The Century Dictionary and Cyclopedia, The Century co., p. 1547.
↑ Trapezoid at Math24.net: Formulas and Tables Archived June 28, 2018, at the Wayback Machine Accessed 1 July 2014.

External links

Some engineering formulas involving isosceles trapezoids

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[1] "Trapezoid - math word definition - Math Open Reference".

[2] Ryoti, Don E. (1967). "What is an Isosceles Trapezoid?". The Mathematics Teacher. 60 (7): 729–730. doi:10.5951/MT.60.7.0729. JSTOR 27957671.

[3] Larson, Ron; Boswell, Laurie (2016). Big Ideas MATH, Geometry, Texas Edition. Big Ideas Learning, LLC (2016). p. 398. ISBN 978-1608408153.

[4] "A Hierarchical Classification of Quadrilaterals". dynamicmathematicslearning.com. Retrieved February 10, 2024.

[esg-5] Halsted, George Bruce (1896), "Chapter XIV. Symmetrical Quadrilaterals", Elementary Synthetic Geometry, J. Wiley & sons, pp. 49–53.

[6] Whitney, William Dwight; Smith, Benjamin Eli (1911), The Century Dictionary and Cyclopedia, The Century co., p. 1547.

[7] Trapezoid at Math24.net: Formulas and Tables Archived June 28, 2018, at the Wayback Machine Accessed 1 July 2014.

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