# Isosceles trapezoid

Isosceles trapezoid | |
---|---|

Isosceles trapezoid with axis of symmetry | |

Type | quadrilateral, trapezoid |

Edges and vertices | 4 |

Symmetry group |
Dih_{2}, [ ], (*), order 2 |

Dual polygon | Kite |

Properties | convex, cyclic |

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. 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). The diagonals are also of 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.

Another special case is a *3-equal side trapezoid*, sometimes known as a *trilateral trapezoid*^{[1]} or a *trisosceles trapezoid*.^{[2]} They can also be seen dissected from regular polygons of 5 sides or more as a truncation of 4 sequential vertices.

The isosceles trapezoid is also rarely known as a *symtra* because of its symmetry.^{[3]}

### Self-intersections

Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite.^{[3]} 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 of an isosceles trapezoid.^{[4]}

Convex isosceles trapezoid |
Crossed isosceles trapezoid |
antiparallelogram |
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## Characterizations

If a quadrilateral is known to be a trapezoid, it is not necessary to check that the legs have the same length in order to know that it is an isosceles trapezoid (nor, under the definitions given in Wikipedia, is it sufficient, 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); any of the following properties also 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).

If rectangles are included in the class of trapezoids then one may concisely define an isosceles trapezoid as "a cyclic quadrilateral with equal diagonals"^{[5]} or as "a cyclic quadrilateral with a pair of parallel sides" or as "a convex quadrilateral with a line of symmetry through the mid-points of opposite sides".

## 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

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,

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

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

The distance from point *E* to base *AD* is given by

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 given as follows:

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

-where 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

## Circumradius

The radius in the circumscribed circle is given by^{[6]}

In a rectangle where *a* = *b* this is simplified to .

## See also

## References

- ↑ Michael de Villiers, Hierarchical Quadrilateral Tree
- ↑ isosceles trapezoid
- 1 2 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. - ↑ Mzone.mweb.co.za
- ↑ Trapezoid at Math24.net: Formulas and Tables Accessed 1 July 2014.