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 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?
 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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geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0Bad drawing, sorry. Hopefully you know what I'm trying to draw...dw:1352593402510:dw

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Do you know any other information?

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0No, this is all given information I have.

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Well it's equilateral so all of it is the same...

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0The area of an equilateral triangle is : \[\frac{ s^2\sqrt{3} }{ 4 }\]

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0And circles are tangent to each other, so they all should be the same.

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Mhmm... Let me see. You post very intrsting questions.

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0And how can I applying to it?

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0Sorry for interruption, but I need your help... @Hero @tcarroll010 @AccessDenied @AriPotta

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0Any ideas, hints, tips?

tcarroll010
 2 years ago
Best ResponseYou've already chosen the best response.1Are all those circles of the same radius?

jon.stromer.galley
 2 years ago
Best ResponseYou've already chosen the best response.2if 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?

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0Well, 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...

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0Go on... @jon.stromer.galley

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Well if they are tengential to each other I assume they have the same radius.

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0So where should we start?

tcarroll010
 2 years ago
Best ResponseYou've already chosen the best response.1I am exploring a relationship between the triangle and a second triangle formed by connecting the centers of the circles.

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0Why not the relationship between the area of circle to the area of triangle? I think this is good start, but I'm not sure.

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0I am thinking about saying that: dw:1352594259221:dw

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0I don't think so though...

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0I doubt it. dw:1352594316081:dw

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah I know. Was just wondering...

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0A better picture perhaps: http://puu.sh/1oLWD

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0I 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.

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0Much better. @AccessDenied

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0Oh 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

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Where are you getting these questions from? O_o .

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Similar triangles perhaps?

jon.stromer.galley
 2 years ago
Best ResponseYou've already chosen the best response.2Ok, 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.

jon.stromer.galley
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1352594743279:dw

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0Hint: you have an equilateral triangle for the exterior, so you know some angles. ;)

tcarroll010
 2 years ago
Best ResponseYou've already chosen the best response.1I'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.

jon.stromer.galley
 2 years ago
Best ResponseYou've already chosen the best response.2let's draw it again...dw:1352594832325:dw

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Isn't that also one of those special triangles?

jon.stromer.galley
 2 years ago
Best ResponseYou've already chosen the best response.2At 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)

tcarroll010
 2 years ago
Best ResponseYou've already chosen the best response.1It looks like a whole bunch of us about simultaneously hit on the right idea! Fun trig at this point.

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0I 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?

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Now we know the radius of all those circles are 1!

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0not necessarily. The 306090 rule only expresses a ratio. :P

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0And from this, we can find the ratio of the total area of six circle to the area of triangle, right?

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Can we not assume? O_o .

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

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Not sure is the side length would just be 2r...

tcarroll010
 2 years ago
Best ResponseYou've already chosen the best response.1To 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!

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Well 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.

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0I am assuming the height would be 3...

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0Now I can't get a ratio....

tcarroll010
 2 years ago
Best ResponseYou've already chosen the best response.1With 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.

Dido525
 2 years ago
Best ResponseYou've already chosen the best response.0So it would just be 2root(3)+2 .

tcarroll010
 2 years ago
Best ResponseYou've already chosen the best response.1If you are going with your assumption or requirement that r is "1", then, yes.

tcarroll010
 2 years ago
Best ResponseYou've already chosen the best response.1So, you should be able to get the altitude using Pythagorean and then area of triangle, and areas of circles. You're just about done!

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

jon.stromer.galley
 2 years ago
Best ResponseYou've already chosen the best response.2or 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...

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

geerky42
 2 years ago
Best ResponseYou've already chosen the best response.0I got the same answer. Thank you, everyone, for your times. I appreciate it.

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.0You're welcome! :)
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