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Math Help - equilateral triangle inscribed in a circle..

  1. #1
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    equilateral triangle inscribed in a circle..

    In the given fig., ABC is an equilateral triangle inscribed in a circle of radius 4 cm. Find the area of the shaded region.
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  2. #2
    A riddle wrapped in an enigma
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    Quote Originally Posted by snigdha View Post
    In the given fig., ABC is an equilateral triangle inscribed in a circle of radius 4 cm. Find the area of the shaded region.
    Hi snigdha,

    The area of the circle is A_c=\pi r^2=16 \pi

    We need to find the area of the triangle, subtract that area from the area of the circle and then divide it by 3.

    Find the area of the equilateral triangle:

    Draw the radius from O to B.
    Draw the perpendicular bisector of BC through O intersecting BC at X.
    Solve the right triangle we just made.

    Hypotenuse = 4
    Since it's a 30-60-90 right triangle, angle OBX = 30 degrees and angle BOX = 60 degrees.

    With a little trig or 30-60-90 rules, we determine that OX = 2, and BX = 2\sqrt{3}.
    Since triangle OBC is isosceles, BX = CX.
    The base of the equilateral triangle ABC is 4\sqrt{3}.

    All we need now is the height of the equilateral triangle.
    We just add the radius 4 to OX and get 6.

    The area of the inscribed equilateral triangle is A_t=\frac{1}{2}bh=\frac{1}{2}(4\sqrt{3})(6)=12\sqr  t{3}.

    Now you have both pieces I talked about in the beginning.
    You may finish up now.
    Turn the lights out when you leave.
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  3. #3
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    You can also look at this: circular segement.
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  4. #4
    A riddle wrapped in an enigma
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    Quote Originally Posted by Plato View Post
    You can also look at this: circular segement.
    I thought about doing it that way, then I thought again.
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  5. #5
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    Well.....i found masters' solution better and way simpler....!
    thanks a lot!!
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