Isosceles trapezoid

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Isosceles trapezoid
Isosceles trapezoid.svg
Isosceles trapezoid with axis of symmetry
Type quadrilateral, trapezoid
Edges and vertices 4
Symmetry group Dih2, [ ], (*), 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, making it automatically a trapezoid. Some sources would qualify this with the exception: "excluding rectangles." Two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (a property shared by the isosceles trapezoid and by 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).

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] The isosceles trapezoid is also (rarely) known as a symtra because of its symmetry.[1]

Special cases[edit]

Examples of isosceles trapezoids (under the inclusive definition of trapezoids) are rectangles and squares.

Characterizations[edit]

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:

  • The diagonals have the same length.
  • The base angles have the same measure.
  • An isosceles triangle is formed by the base and the extensions of the legs. (Rectangles are excluded here.)
  • 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 AECE 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"[3] or as "a cyclic quadrilateral with a pair of parallel sides."

Angles[edit]

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[edit]

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[edit]

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+b\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 + b + 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+b)^2(a-b+2c)(b-a+2c)}{16}}.

Circumradius[edit]

The radius in the circumscribed circle is given by[4]

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}.

See also[edit]

References[edit]

  1. ^ a b Halsted, George Bruce (1896), "Chapter XIV. Symmetrical Quadrilaterals", Elementary Synthetic Geometry, J. Wiley & sons, pp. 49–53 .
  2. ^ Whitney, William Dwight; Smith, Benjamin Eli (1911), The Century Dictionary and Cyclopedia, The Century co., p. 1547 .
  3. ^ Mzone.mweb.co.za
  4. ^ Trapezoid at Math24.net: Formulas and Tables [1] Accessed 1 July 2014.

External links[edit]