Isosceles trapezoid

In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English, also known as a symtra[1]) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid. Some sources would qualify all this with the exception: "excluding rectangles." Two opposite sides (bases) are parallel, the two other sides (legs) are of equal length. The diagonals are of equal length. An isosceles trapezoid's base angles are equal in measure.

Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite.[1] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also 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.[2]

Contents

Special cases

Examples of isosceles trapezoids are rectangles and squares.

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; any of the following properties also distinguishes an isosceles trapezoid from other trapezoids:

If rectangles are included in the class of trapezoids then one may concisely define an isosceles trapezoid as "a cyclic quadrilateral with equal diagonals"[3] or as "a cyclic quadrilateral with a pair of parallel sides."

Angles

In an isosceles trapezoid the base angles have the same measure pairwise. In the picture on the right, angles \angle ABC and \angle DCB are obtuse angles of the same measure, while angles \angle BAD and \angle 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 \angle ABC %2B \angle BAD = 180^\circ.

Diagonals and height

The diagonals of an isosceles trapezoid have the same length and divide each other into segments of the same length. In the picture below, the diagonals AC and BD have the same length, that is AC = BD; and they divide each other in segments of the same length, that is, 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%2Bc^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%2Bb}{2}\right)^2}=\tfrac{1}{2}\sqrt{4c^2-(a-b)^2}.

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 diagram to the right, 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

K=\frac{h\left(a%2Bb\right)}{2}.

If instead of the height of the trapezoid, the length of the leg AB = c is known, then the area can be computed using the formula

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

where s = \tfrac{1}{2}(a %2B b %2B 2c) is the semiperimeter 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= \sqrt{\frac{(a%2Bb)^2(a-b%2B2c)(b-a%2B2c)}{16}}.

Circumradius

The radius in the circumcribed circle is given by

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

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

See also

References

  1. ^ a b Halsted, George Bruce (1896), "Chapter XIV. Symmetrical Quadrilaterals", Elementary Synthetic Geometry, J. Wiley & sons, pp. 49–53, http://books.google.com/books?id=H3ALAAAAYAAJ&pg=PA49 .
  2. ^ Whitney, William Dwight; Smith, Benjamin Eli (1911), The Century Dictionary and Cyclopedia, The Century co., p. 1547, http://books.google.com/books?id=ownpAAAAMAAJ&pg=PA1547 .
  3. ^ Mzone.mweb.co.za

External links