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

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

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

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

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

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

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

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

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


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

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


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