Isosceles Trapezoid
GeometryAn isosceles trapezoid is a trapezoid with equal legs, equal base angles, and congruent diagonals.
Formula
\text{Area} = (1/2)(b_1 + b_2) \times \text{height}
Definition
An isosceles trapezoid is a trapezoid whose two non-parallel sides (legs) are equal in length. It is symmetrical: the left and right sides are mirror images of each other.
Example
The shape of a lampshade or an upside-down bucket with equal-length slanted sides is an isosceles trapezoid. If you fold it in half down the middle, both halves match exactly.
Key Insight
Like an isosceles triangle, an isosceles trapezoid has one line of symmetry - the perpendicular bisector of the two bases. The two base angles on each base are equal, just like the equal angles at the base of an isosceles triangle.
Definition
An isosceles trapezoid is a trapezoid with congruent legs (non-parallel sides). Properties: base angles are congruent (both lower base angles equal, both upper base angles equal), diagonals are congruent, and it is a cyclic quadrilateral (can be inscribed in a circle). It has one line of symmetry.
Example
Isosceles trapezoid with bases $10$ and $6$ and legs $5$: lower base angles are equal. Height $= \sqrt{5^2 - 2^2} = \sqrt{21}$. Area $= (1/2)(16)(\sqrt{21}) = 8\sqrt{21}$ approximately $36.7$.
Key Insight
An isosceles trapezoid is always a cyclic quadrilateral. This is because its opposite angles are supplementary: the equal base angles plus the equal upper angles sum to $180$. This makes it possible to draw a circle through all four vertices.
Definition
An isosceles trapezoid has a line of reflective symmetry (the perpendicular bisector of both bases). It is cyclic: opposite angles are supplementary, so it can be inscribed in a circle. Its diagonals are equal (can be proved by the symmetry: the reflection maps each diagonal to the other, preserving length). The symmetry group is $\mathbb{Z}_2$.
Example
Isosceles trapezoid with bases $b_1, b_2$ and legs $a$. Diagonal length: $d = \sqrt{a^2 + b_1 b_2}$. For $b_1=10$, $b_2=4$, $a=5$: $d = \sqrt{25+40} = \sqrt{65}$.
Key Insight
The fact that an isosceles trapezoid is cyclic connects it to Ptolemy's theorem: for the isosceles trapezoid $ABCD$ with diagonals $AC = BD = d$, and bases $AB = b_1$ and $CD = b_2$: Ptolemy gives $d^2 = b_1 b_2 + a^2$, confirming the diagonal formula. Ptolemy's theorem unifies both the Pythagorean theorem (rectangle case) and this isosceles trapezoid result.