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Thread: Equilateral Triangle

  1. #1
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    Equilateral Triangle

    Find the perimeter of the equilateral triangle inscribed in a circle of radius 20.0 inches
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  2. #2
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    Hello, magentarita!

    Find the perimeter of the equilateral triangle inscribed in a circle of radius 20 inches
    Code:
                    A
                  * * *
              *    /|\    *
            *     / | \     *
           *     /  |  \     *
                / 20|   \  D
          *    /    |    *    *
          *   /     *     \   *
          *  /      O      \  *
            /               \
         B *- - - - - - - - -* C
            *               *
              *           *
                  * * *

    The triangle is $\displaystyle ABC$
    The circle has center $\displaystyle O$ and radius $\displaystyle OA = 20$
    Draw $\displaystyle OD \perp AC.$

    Since $\displaystyle \Delta ABC$ is equilateral, $\displaystyle \angle A \:=\:\angle B \:=\:\angle C \:=\:60^o$
    . . Then: $\displaystyle \angle OAD = 30^o$

    In right triangle $\displaystyle ADO\!:\;\;\cos30^o \:=\:\frac{AD}{20} \quad\Rightarrow\quad AD \:=\:20\cos30^o \:\approx\:17.32$ inches

    The side of the triangle is: .$\displaystyle 2\times 17.32 \:=\:34.64$ inches


    Therefore, the perimeter is: .$\displaystyle 3 \times 34.64 \:=\:103.92$ inches.

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  3. #3
    MHF Contributor
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    wonderfully done.........

    Quote Originally Posted by Soroban View Post
    Hello, magentarita!

    Code:
                    A
                  * * *
              *    /|\    *
            *     / | \     *
           *     /  |  \     *
                / 20|   \  D
          *    /    |    *    *
          *   /     *     \   *
          *  /      O      \  *
            /               \
         B *- - - - - - - - -* C
            *               *
              *           *
                  * * *
    The triangle is $\displaystyle ABC$
    The circle has center $\displaystyle O$ and radius $\displaystyle OA = 20$
    Draw $\displaystyle OD \perp AC.$

    Since $\displaystyle \Delta ABC$ is equilateral, $\displaystyle \angle A \:=\:\angle B \:=\:\angle C \:=\:60^o$
    . . Then: $\displaystyle \angle OAD = 30^o$

    In right triangle $\displaystyle ADO\!:\;\;\cos30^o \:=\:\frac{AD}{20} \quad\Rightarrow\quad AD \:=\:20\cos30^o \:\approx\:17.32$ inches

    The side of the triangle is: .$\displaystyle 2\times 17.32 \:=\:34.64$ inches


    Therefore, the perimeter is: .$\displaystyle 3 \times 34.64 \:=\:103.92$ inches.
    It's amazing how you used trig to answer a geometry question.
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