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Math Help - Circle inscribed in an equlitaeral triangle

  1. #1
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    Circle inscribed in an equlitaeral triangle

    A circle of radius 4 is inscribed within an equilateral triangle. Find the area of the triangle.

    I am unsure on how to find the height of the triangle and the base.
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  2. #2
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    Re: Circle inscribed in an equlitaeral triangle

    in circle radius =4
    circumcircle radius = 8
    length of altitude =12
    You do the rest using the properties of 30-60-90 triangles
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    Re: Circle inscribed in an equlitaeral triangle

    If an equilateral triangle has side length s, then any altitude divides it into two right triangles with hypotenuse of length s and one leg of length s/2. By the pythagorean theorem, the other leg, the altitude of the equilateral triangle has length b given by b^2+ \frac{s^2}{4}= s^2 so that b^2= s^2- \frac{s^2}{4}= \frac{3s^2}{4} so that b= \frac{\sqrt{3}}{2}s.

    Of course all three altitudes cross at a single point. Let x be the distance from the foot of one altitude to that point so the distance from that point to a vertex is \frac{\sqrt{3}}{2}s- x. Then we have a right triangle whose vertices are that point of intersection, the foot of an altitude, and one vertex on that base. That right triangle has one leg of length s/2, another leg of length x, and hypotenuse of length \frac{\sqrt{3}}{2}s- x. Put those into the Pythagorean theorem to find x, the radius of the circle, as a function of s.
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    Re: Circle inscribed in an equlitaeral triangle

    Hello, johnsy123!

    Here is another solution, using some clever formulas.


    A circle of radius 4 is inscribed within an equilateral triangle.
    Find the area of the triangle.
    Code:
                          A
                          *
                         / \
                        /   \
                       /     \
                      /       \
                     /         \
                    /   * * *   \
                   /*           *\
                  *               *
                 *                 *
                /                   \
               /*                   *\
              / *         *         * \
             /  *         |         *  \
            /             |             \
           /     *        |4       *     \
          /       *       |       *       \
         /          *     |     *          \
      B *---------------*-*-*---------------* C
        : - - - - - - - - x - - - - - - - - :
    The area of an equilateral triangle with side x is: . A \:=\:\tfrac{\sqrt{3}}{4}x^2 .[1]

    The area of a triangle is given by: . A \:=\:\tfrac{1}{2}pr
    . . where p is the perimeter and r is the radius of the inscribed circle.
    So we have: . A \:=\:\tfrac{1}{2}(3x)(4) \:=\:6x .[2]

    Equate [1] and [2]: . \tfrac{\sqrt{3}}{4}x^2 \:=\:6x \quad\Rightarrow\quad \sqrt{3}x^2 \:=\:24x

    Since x \ne 0, divide by x\!:\;\sqrt{3}x \:=\:24 \quad\Rightarrow\quad x \:=\:\tfrac{24}{\sqrt{3}} \:=\:8\sqrt{3}

    Substitute into [1]: . A \;=\;\tfrac{\sqrt{3}}{4}(8\sqrt{3})^2 \;=\;48\sqrt{3}
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    Re: Circle inscribed in an equlitaeral triangle

    I don't understand how you got the area of the equilateral triangle to be A= SQRT{3}X^2/4.......
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    Re: Circle inscribed in an equlitaeral triangle

    How do you prove that the altitude is 12? you can't really just say that the distance from the tip of the circle to the sharp point of the triangle is the same as the radius. A proven statement would better clarify my understanding.
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  7. #7
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    Re: Circle inscribed in an equlitaeral triangle

    Hi johnsey123,
    Draw an equilateral triangle and its three medians.These are perpendicular to each side.Note that you have created 6 triangles which are congruent to each other.Each one is a 30-60- 90 triangle with one side given (4).(incircle radius)The alttude has another segment which you derive from the property of 30-60-90 triangle. The ratio of its sides is 2-1-rad3 so the hyp is 2 times 4= 8 making the altitude 4+8 =12.The other leg of this triangle is 4rad3= 1/2 of the side of the equil tria so the area of the equilateral tri is 12 *4rad3 = 48rad3. You could also calculate the area by caculating the area of the small tri and multiplying by 6
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  8. #8
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    Re: Circle inscribed in an equlitaeral triangle

    Can you please construct a diagram to show this? I think i may have constructed one but i am not so sure if it is correct.
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  9. #9
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    Re: Circle inscribed in an equlitaeral triangle

    Go to post 4 and finish adding lines to the drawing there as I described
    Last edited by bjhopper; June 3rd 2012 at 07:14 PM. Reason: change wording
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