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Elementary Geometry

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Trapezoid

  • Bases of a trapezoid: \(a\), \(b\)
    Legs of a trapezoid: \(c\), \(d\)
    Midline of a trapezoid: \(m\)
    Altitude of a trapezoid: \(h\)
    Perimeter: \(P\)
    Diagonals of a trapezoid: \(p\), \(q\)
    Angle between the diagonals: \(\varphi\)
    Radius of the circumscribed circle: \(R\)
    Radius of the inscribed circle: \(r\)
    Area: \(S\)
    1. A trapezoid (or a trapezium) is a quadrilateral in which (at least) one pair of opposite sides is parallel. Sometimes a trapezoid is defined as a quadrilateral having exactly one pair of parallel sides. The parallel sides are called the bases, and two other sides are called the legs.
    2. Trapezoid
    3. A trapezoid in which the legs are equal is called an isosceles trapezoid. A trapezoid in which at least one angle is the right angle (\(90^\circ\)) is called a right trapezoid.
    4. The midline of a trapezoid is parallel to the bases and equal to the arithmetic mean of the lengths of the bases.
      \(m = {\large\frac{{a + b}}{2}\normalsize},\;\) \(m\parallel a,\;\) \(m\parallel b\)
    5. Diagonals of a trapezoid (if \(a \gt b\))
      \(p = \sqrt {\large\frac{{{a^2}b – a{b^2} – b{c^2} + a{d^2}}}{{a – b}}\normalsize},\;\) \(q = \sqrt {\large\frac{{{a^2}b – a{b^2} – b{d^2} + a{c^2}}}{{a – b}}\normalsize} \)
    6. Perimeter of a trapezoid
      \(P = a + b + c + d\)
    7. Area of a trapezoid
      \(S = {\large\frac{{a + b}}{2}\normalsize} h = mh\)
      \(S =\) \({\large\frac{{a + b}}{2}\normalsize} \) \(\sqrt {{c^2} – {{\left[ {\large\frac{{{{\left( {a – b} \right)}^2} + {c^2} – {d^2}}}{{2\left( {a – b} \right)}}\normalsize} \right]}^2}} \)
    8. All four vertices of an isosceles trapezoid lie on a circumscribed circle.
    9. The circle circumscribed about an isosceles trapezoid
    10. Radius of the circle circumscribed about an isosceles trapezoid
      \(R =\) \({\large\frac{{c\sqrt {ab + {c^2}} }}{{\sqrt {\left( {2c – a + b} \right)\left( {2c + a – b} \right)} }}\normalsize}\)
    11. Diagonal of an isosceles trapezoid
      \(p = \sqrt {ab + {c^2}} \)
    12. Altitude of an isosceles trapezoid
      \(h = \sqrt {{c^2} – {\large\frac{1}{4}\normalsize}{{\left( {a – b} \right)}^2}} \)
    13. If the sum of the bases of a trapezoid is equal to the sum of its legs, all four sides of the trapezoid are tangents to an inscribed circle:
      \(a + b = c + d\)
    14. The circle inscribed in a trapezoid
    15. Radius of the inscribed circle
      \(r = {\large\frac{h}{2}\normalsize},\)
      where \(h\) is the altitude of the trapezoid.