Elementary Geometry

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  • Side of a rhombus: \(a\)
    Diagonals of a rhombus: \({d_1}\), \({d_2}\)
    Consecutive angles: \(\alpha\), \(\beta\)
    Altitude: \(h\)
    Radius of the inscribed circle: \(r\)
    Perimeter: \(P\)
    Area: \(S\)
    1. A rhombus is a parallelogram in which all four sides are equal.
    2. Rhombus
    3. The sum of the angles adjacent to any side of a rhombus is \(180^\circ:\)
      \(\alpha + \beta = 180^\circ\)
    4. The diagonals of a rhombus are perpendicular and bisect each other.
    5. If a rhombus has one right angle it is a square.
    6. Relation between the sides and the diagonals of rhombus
      \(d_1^2 + d_2^2 = 4{a^2}\)
    7. Altitude of a rhombus
      \(h = a\sin \alpha =\) \({\large\frac{{{d_1}{d_2}}}{{2a}}\normalsize}\)
    8. Radius of the inscribed circle
      \(r = {\large\frac{h}{2}\normalsize} = {\large\frac{{a\sin \alpha }}{2}\normalsize} =\) \({\large\frac{{{d_1}{d_2}}}{{4a}}\normalsize} =\) \({\large\frac{{{d_1}{d_2}}}{{2\sqrt {d_1^2 + d_2^2} }}\normalsize}\)
    9. Perimeter of a rhombus
      \(P = 4a\)
    10. Area of a rhombus
      \(S = ah =\) \({a^2}\sin \alpha =\) \({\large\frac{{{d_1}{d_2}}}{2}\normalsize}\)