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Math Help - Rhombus

  1. #1
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    Question Rhombus

    The sides of a rhombus are 10cm long. If the lengths of the diagonals differ by 4, what is the area of the rhombus?
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  2. #2
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    Hello, jumpman23!

    The sides of a rhombus are 10cm long.
    If the lengths of the diagonals differ by 4, what is the area of the rhombus?

    You're expected to know a few facts about a rhombus.

    . . The diagonals of a rhombus are perpendicular.

    . . If the diagonals are d_1 and d_2, the area is: . A \:=\:\frac{1}{2}d_1d_2
    Code:
                A     10    B 
                * - - - - - *
               / *       * /
              /   *   *   /
           10/     *O    / 10
            /   *   *   /
           / *       * /
          * - - - - - *
          D    10     C

    We have: . AB \,=\, BC \,=\, BD \,=\, DA \,=\, 10

    Let AC\,= \,d. .Then BD\,=\,d + 4

    Then: . AO \,= \,\frac{d}{2},\;BO \,= \,\frac{d+4}{2}

    In right triangle AOB :\;\;AO^2 + BO^2\:=\:AB^2

    . . so we have: . \left(\frac{d}{2}\right)^2 + \left(\frac{d+4}{2}\right)^2\;=\;10^2

    . . which simplifies to: . d^2 + 4d - 192\:=\:0

    . . which factors: . (d -12)(d + 16)\:=\:0

    . . and has roots: . d \:=\:12,\,-16


    The diagonals are: . AC \,=\, d \,= \,12,\;BD \,= \,d+4 \,= \,16

    Therefore, the area is: . A \:=\:\frac{1}{2}(12)(16)\:=\:96 cm².


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