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Math Help - Plane Geometry Basics

  1. #1
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    Plane Geometry Basics

    Hi there!

    currently having trouble with a problem found on page 8 #18 in Kern and Bland Solid Mensuration



    The problem states that

    "Each of the four circles shown in the figure above is tangent to the other three.
    (a) If the radius of each of the smaller circles is a, find the area of the largest circle.
    (b) If a = 2.71 what is the area of the largest circle?"


    I know that I need to form a triangle from the radii of the 3 circles but the rest is a blank to me.

    I also need to show the solution that was shown in the book 1/2 (7 + 4sqrt3)a2 (I think? I can't see the solution clearly since its an old book)



    Thank you very much



    Edit: Solved it before...but I need to use that specific solution shown in the book...

    MY instructor said so...
    Last edited by keosan22; July 9th 2013 at 05:12 PM.
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  2. #2
    MHF Contributor
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    Re: Plane Geometry Basics

    Hey keosan22.

    Hint: Try forming an equilateral triangle using the three points of each circle at its tangent point with the large circle and then calculate the center of this triangle to get the centre of the big circle and then its radius as a function of a.

    You will have to prove that the triangle is equilateral, and depending on what knowledge you have, I would suggest using a symmetry argument of some sort.
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  3. #3
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    Re: Plane Geometry Basics

    Plane Geometry Basics-10-jul-13.png
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  4. #4
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    Re: Plane Geometry Basics

    Thank you both for your replies.

    I am grateful that people still answer when asked

    I stopped studying for a long time...many reasons...poverty in the Philippines is one of them
    Things tend to slip after a few years. I am humbled by my inability to answer this question...and mortified (I am really gonna need a lot of help to graduate)

    My fellow students and instructors have quite a busy schedule so I couldn't get around to asking them

    Gonna try to solve the ones I can on my own...when I stumble across something I can't I'll post it here and hopefully someone replies


    Thank you for your time

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  5. #5
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    Re: Plane Geometry Basics

    Thanks for such that help me also in understanding the plane geometry basics.. This post clear all my doubt regarding this topic... Now i am very much satisfy with my concept...
    Thanks again...
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  6. #6
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    Re: Plane Geometry Basics

    this may help
    draw an equilateral triangle and its three medians
    small circle radius a
    medians meet in centroid of triangle which is center of large circle
    use properties of 30-60-90 triangle to find the distance between the centroid and a triangle vertex
    radius of large circle is that distance plus a
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