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geerky42

  • 2 years ago

Six circles are tangent to each other and an equilateral triangle is inscribed around them as shown. What percent of the area of triangle is NOT shaded?

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  1. geerky42
    • 2 years ago
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    Bad drawing, sorry. Hopefully you know what I'm trying to draw...|dw:1352593402510:dw|

  2. Dido525
    • 2 years ago
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    Do you know any other information?

  3. geerky42
    • 2 years ago
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    No, this is all given information I have.

  4. Dido525
    • 2 years ago
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    Well it's equilateral so all of it is the same...

  5. geerky42
    • 2 years ago
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    I guess so.

  6. Dido525
    • 2 years ago
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    The area of an equilateral triangle is : \[\frac{ s^2\sqrt{3} }{ 4 }\]

  7. geerky42
    • 2 years ago
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    And circles are tangent to each other, so they all should be the same.

  8. Dido525
    • 2 years ago
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    Mhmm... Let me see. You post very intrsting questions.

  9. geerky42
    • 2 years ago
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    And how can I applying to it?

  10. Dido525
    • 2 years ago
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    I am thinking...

  11. geerky42
    • 2 years ago
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    Sorry for interruption, but I need your help... @Hero @tcarroll010 @AccessDenied @AriPotta

  12. geerky42
    • 2 years ago
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    Any ideas, hints, tips?

  13. tcarroll010
    • 2 years ago
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    Are all those circles of the same radius?

  14. jon.stromer.galley
    • 2 years ago
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    if you know the area of the triangle and the area of the circle(s) what can you say about the area not covered by the circles?

  15. geerky42
    • 2 years ago
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    Well, we only know that they are tangent to each other and in the shown image on my paper, they appear that they also tangent to the sides of triangle too, so I guess yeah. This is literally the only given information I have...

  16. geerky42
    • 2 years ago
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    Go on... @jon.stromer.galley

  17. Dido525
    • 2 years ago
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    Well if they are tengential to each other I assume they have the same radius.

  18. geerky42
    • 2 years ago
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    I suppose.

  19. geerky42
    • 2 years ago
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    So where should we start?

  20. tcarroll010
    • 2 years ago
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    I am exploring a relationship between the triangle and a second triangle formed by connecting the centers of the circles.

  21. geerky42
    • 2 years ago
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    Why not the relationship between the area of circle to the area of triangle? I think this is good start, but I'm not sure.

  22. geerky42
    • 2 years ago
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    I don't know, lol.

  23. Dido525
    • 2 years ago
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    I am thinking about saying that: |dw:1352594259221:dw|

  24. Dido525
    • 2 years ago
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    I don't think so though...

  25. geerky42
    • 2 years ago
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    I doubt it. |dw:1352594316081:dw|

  26. Dido525
    • 2 years ago
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    Yeah I know. Was just wondering...

  27. AccessDenied
    • 2 years ago
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    A better picture perhaps: http://puu.sh/1oLWD

  28. geerky42
    • 2 years ago
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    I think @tcarroll010 has a good point, perhaps we should determine the relationship between the triangle and a second triangle formed by connecting the centers of the circles.

  29. geerky42
    • 2 years ago
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    Much better. @AccessDenied

  30. Dido525
    • 2 years ago
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    LOT better.

  31. AccessDenied
    • 2 years ago
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    Oh wow, this is sort of interesting: I draw in all of the tangents of the circles to the exterior triangle and an interior triangle: http://puu.sh/1oM08

  32. Dido525
    • 2 years ago
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    Where are you getting these questions from? O_o .

  33. Dido525
    • 2 years ago
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    Hmm!

  34. Dido525
    • 2 years ago
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    Similar triangles perhaps?

  35. jon.stromer.galley
    • 2 years ago
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    Ok, so all you know for sure is that the triangle is equilateral and that the circles have a radius r. That is enough to express the combined circle area, but to answer the question you need to know how big the triangle is (how long the sides are) in terms of R. Let's look at the lower left corner.

  36. jon.stromer.galley
    • 2 years ago
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    |dw:1352594743279:dw|

  37. Dido525
    • 2 years ago
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    We don't know that...

  38. AccessDenied
    • 2 years ago
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    Hint: you have an equilateral triangle for the exterior, so you know some angles. ;)

  39. tcarroll010
    • 2 years ago
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    I've got what might be a good idea. You use trig. And 30-60-90 right triangles in the corners. I was independently working a similar diagram to Accessd.

  40. jon.stromer.galley
    • 2 years ago
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    let's draw it again...|dw:1352594832325:dw|

  41. Dido525
    • 2 years ago
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    Ohh yeah!!!

  42. Dido525
    • 2 years ago
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    Isn't that also one of those special triangles?

  43. jon.stromer.galley
    • 2 years ago
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    At this point if you have been shown you have enough data to figure out the base of the triangle. The length of a side of the over all triangle will be then 3r + 2( length of the base of the triangle we just drew)

  44. tcarroll010
    • 2 years ago
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    It looks like a whole bunch of us about simultaneously hit on the right idea! Fun trig at this point.

  45. geerky42
    • 2 years ago
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    I have sense I'm getting closer to solution, yet I have no idea where to start. I have to find the relationship between the radius and length of triangle side, right?

  46. Dido525
    • 2 years ago
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    |dw:1352594992142:dw|

  47. Dido525
    • 2 years ago
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    Now we know the radius of all those circles are 1!

  48. AccessDenied
    • 2 years ago
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    not necessarily. The 30-60-90 rule only expresses a ratio. :P

  49. geerky42
    • 2 years ago
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    And from this, we can find the ratio of the total area of six circle to the area of triangle, right?

  50. Dido525
    • 2 years ago
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    Can we not assume? O_o .

  51. jon.stromer.galley
    • 2 years ago
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    the radius is still "1" r but now the base can be expressed in terms of r as well

  52. Dido525
    • 2 years ago
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    Not sure is the side length would just be 2r...

  53. tcarroll010
    • 2 years ago
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    To be pure, you would use "r" for radius, but you are free to call radius "1" for your purposes, so, yes you can make the assumption, with that proviso. And then it's simple to get your triangle side lengths. Nice little problem!

  54. Dido525
    • 2 years ago
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    Well looking at @AccessDenied Drawing... : Since the radius is 1 we can say the side length's of the smaller, inside triangle are 3, 3 and 2 going clockwise.

  55. Dido525
    • 2 years ago
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    4 4 and 4 sorry.

  56. Dido525
    • 2 years ago
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    |dw:1352595331990:dw|

  57. Dido525
    • 2 years ago
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    I am assuming the height would be 3...

  58. Dido525
    • 2 years ago
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    |dw:1352595405601:dw|

  59. Dido525
    • 2 years ago
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    Now I can't get a ratio....

  60. tcarroll010
    • 2 years ago
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    With a circle radius of "r", being the "short leg", then the "long leg" will be r times sqrt(3). So a triangle side will be 2r(sqrt(3)) + 2r.

  61. Dido525
    • 2 years ago
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    So it would just be 2root(3)+2 .

  62. tcarroll010
    • 2 years ago
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    If you are going with your assumption or requirement that r is "1", then, yes.

  63. tcarroll010
    • 2 years ago
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    So, you should be able to get the altitude using Pythagorean and then area of triangle, and areas of circles. You're just about done!

  64. jon.stromer.galley
    • 2 years ago
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    I calculate that the length of a side of the master triangle is 6.4641r Anyone else? tanks to: adjacent = r / tan(30)

  65. jon.stromer.galley
    • 2 years ago
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    or alternatively by expressing the length of the line as: 2r * root(3) + 3r that plugs in nicely to the formula for the area of an el triangle: (length of side * root(3)) / 4 so that tells us the area is...

  66. AccessDenied
    • 2 years ago
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    I missed the answer placed by the original poster, but my work in ms paint is here: http://puu.sh/1oNgt

  67. geerky42
    • 2 years ago
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    I got the same answer. Thank you, everyone, for your times. I appreciate it.

  68. AccessDenied
    • 2 years ago
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    You're welcome! :)

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