Here's the question you clicked on:
geerky42
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?
Bad drawing, sorry. Hopefully you know what I'm trying to draw...|dw:1352593402510:dw|
Do you know any other information?
No, this is all given information I have.
Well it's equilateral so all of it is the same...
The area of an equilateral triangle is : \[\frac{ s^2\sqrt{3} }{ 4 }\]
And circles are tangent to each other, so they all should be the same.
Mhmm... Let me see. You post very intrsting questions.
And how can I applying to it?
Sorry for interruption, but I need your help... @Hero @tcarroll010 @AccessDenied @AriPotta
Any ideas, hints, tips?
Are all those circles of the same radius?
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?
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...
Go on... @jon.stromer.galley
Well if they are tengential to each other I assume they have the same radius.
So where should we start?
I am exploring a relationship between the triangle and a second triangle formed by connecting the centers of the circles.
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.
I am thinking about saying that: |dw:1352594259221:dw|
I don't think so though...
I doubt it. |dw:1352594316081:dw|
Yeah I know. Was just wondering...
A better picture perhaps: http://puu.sh/1oLWD
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.
Much better. @AccessDenied
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
Where are you getting these questions from? O_o .
Similar triangles perhaps?
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.
|dw:1352594743279:dw|
Hint: you have an equilateral triangle for the exterior, so you know some angles. ;)
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.
let's draw it again...|dw:1352594832325:dw|
Isn't that also one of those special triangles?
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)
It looks like a whole bunch of us about simultaneously hit on the right idea! Fun trig at this point.
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?
Now we know the radius of all those circles are 1!
not necessarily. The 30-60-90 rule only expresses a ratio. :P
And from this, we can find the ratio of the total area of six circle to the area of triangle, right?
Can we not assume? O_o .
the radius is still "1" r but now the base can be expressed in terms of r as well
Not sure is the side length would just be 2r...
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!
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.
I am assuming the height would be 3...
Now I can't get a ratio....
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.
So it would just be 2root(3)+2 .
If you are going with your assumption or requirement that r is "1", then, yes.
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!
I calculate that the length of a side of the master triangle is 6.4641r Anyone else? tanks to: adjacent = r / tan(30)
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...
I missed the answer placed by the original poster, but my work in ms paint is here: http://puu.sh/1oNgt
I got the same answer. Thank you, everyone, for your times. I appreciate it.
You're welcome! :)