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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?
 one year ago
 one year 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?
 one year ago
 one year ago

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geerky42Best ResponseYou've already chosen the best response.0
Bad drawing, sorry. Hopefully you know what I'm trying to draw...dw:1352593402510:dw
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Do you know any other information?
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
No, this is all given information I have.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Well it's equilateral so all of it is the same...
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
The area of an equilateral triangle is : \[\frac{ s^2\sqrt{3} }{ 4 }\]
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
And circles are tangent to each other, so they all should be the same.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Mhmm... Let me see. You post very intrsting questions.
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
And how can I applying to it?
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
Sorry for interruption, but I need your help... @Hero @tcarroll010 @AccessDenied @AriPotta
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
Any ideas, hints, tips?
 one year ago

tcarroll010Best ResponseYou've already chosen the best response.1
Are all those circles of the same radius?
 one year ago

jon.stromer.galleyBest ResponseYou've already chosen the best response.2
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?
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
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...
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
Go on... @jon.stromer.galley
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Well if they are tengential to each other I assume they have the same radius.
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
So where should we start?
 one year ago

tcarroll010Best ResponseYou've already chosen the best response.1
I am exploring a relationship between the triangle and a second triangle formed by connecting the centers of the circles.
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
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.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
I am thinking about saying that: dw:1352594259221:dw
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
I don't think so though...
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
I doubt it. dw:1352594316081:dw
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Yeah I know. Was just wondering...
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
A better picture perhaps: http://puu.sh/1oLWD
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
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.
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
Much better. @AccessDenied
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
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
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Where are you getting these questions from? O_o .
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Similar triangles perhaps?
 one year ago

jon.stromer.galleyBest ResponseYou've already chosen the best response.2
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.
 one year ago

jon.stromer.galleyBest ResponseYou've already chosen the best response.2
dw:1352594743279:dw
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
Hint: you have an equilateral triangle for the exterior, so you know some angles. ;)
 one year ago

tcarroll010Best ResponseYou've already chosen the best response.1
I've got what might be a good idea. You use trig. And 306090 right triangles in the corners. I was independently working a similar diagram to Accessd.
 one year ago

jon.stromer.galleyBest ResponseYou've already chosen the best response.2
let's draw it again...dw:1352594832325:dw
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Isn't that also one of those special triangles?
 one year ago

jon.stromer.galleyBest ResponseYou've already chosen the best response.2
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)
 one year ago

tcarroll010Best ResponseYou've already chosen the best response.1
It looks like a whole bunch of us about simultaneously hit on the right idea! Fun trig at this point.
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
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?
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Now we know the radius of all those circles are 1!
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
not necessarily. The 306090 rule only expresses a ratio. :P
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
And from this, we can find the ratio of the total area of six circle to the area of triangle, right?
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Can we not assume? O_o .
 one year ago

jon.stromer.galleyBest ResponseYou've already chosen the best response.2
the radius is still "1" r but now the base can be expressed in terms of r as well
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Not sure is the side length would just be 2r...
 one year ago

tcarroll010Best ResponseYou've already chosen the best response.1
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!
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
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.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
I am assuming the height would be 3...
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
Now I can't get a ratio....
 one year ago

tcarroll010Best ResponseYou've already chosen the best response.1
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.
 one year ago

Dido525Best ResponseYou've already chosen the best response.0
So it would just be 2root(3)+2 .
 one year ago

tcarroll010Best ResponseYou've already chosen the best response.1
If you are going with your assumption or requirement that r is "1", then, yes.
 one year ago

tcarroll010Best ResponseYou've already chosen the best response.1
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!
 one year ago

jon.stromer.galleyBest ResponseYou've already chosen the best response.2
I calculate that the length of a side of the master triangle is 6.4641r Anyone else? tanks to: adjacent = r / tan(30)
 one year ago

jon.stromer.galleyBest ResponseYou've already chosen the best response.2
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...
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
I missed the answer placed by the original poster, but my work in ms paint is here: http://puu.sh/1oNgt
 one year ago

geerky42Best ResponseYou've already chosen the best response.0
I got the same answer. Thank you, everyone, for your times. I appreciate it.
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.0
You're welcome! :)
 one year ago
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