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
    \(S = ah =\) \({a^2}\sin \alpha =\) \({\large\frac{{{d_1}{d_2}}}{2}\normalsize}\)