Trapezoids

Definition

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

• Bases: The two parallel sides.
• Legs: The two non-parallel sides.
• Base Angles: Two angles that share a base as a common side. (A trapezoid has two pairs of base angles).

Why It Matters

The trapezoid is a fundamental building block in both geometry and numerical analysis. The ‘Trapezoidal Rule’ is the basis for most simple numerical integration, allowing computers to estimate areas and volumes that lack a simple algebraic solution.

Core Concepts

• Isosceles Trapezoid: A trapezoid where the legs are congruent.
  • Properties:
    • Both pairs of base angles are congruent.
    • The diagonals are congruent.
• Median (Midsegment) of a Trapezoid: The segment joining the midpoints of the legs.
  • Median Theorem: The median is parallel to the bases and its length ($m$) is the average of the base lengths ($b_1, b_2$): $m = \frac{b_1 + b_2}{2}$
    • How to read: “m equals b-one plus b-two, over two.”
    • Meaning: The midsegment length is the arithmetic mean of the two parallel bases—the “average width” of the trapezoid.
• Altitude: A perpendicular segment from one base to the other.

Connected Concepts

• Midsegment Theorem
• Parallelograms
• Area of a Triangle