[bonanova]

You have 5 circular disks all the same diameter and equally spaced horizontally and vertically.

Make a dot in the center of the red disk.

Draw a line segment through this dot that divides the 5 disks into two equal areas.

[Yoruichi-san]

that goes from the point to a point that is 1/3 the distance between the center of the adjacent circle to the center of the circle above it, or put mathematically, a line that goes at arctan(1/3) from the x-axis.

[Yoruichi-san]

that goes from the point to a point that is 1/3 the distance between the center of the adjacent circle to the center of the circle above it, or put mathematically, a line that goes at arctan(1/3) from the x-axis.

I have no skill with draw software, so I'll just try to explain where I got this from in words:

A line that goes through the center of the red circle will always cut the circle in half, so the problem is really how to cutting the other four into two parts that each contain a total area of 2 circles.

If I draw line segments between the centers of all four black circles I make a square. I call the side of this square s. If I draw another line segment between the center of the red circle to the black circle adjacent to it, the length of this segment is also s.

Now I need to draw a line that passes through the red dot and cuts the square in half. I call the distance from the center of the lower left black circle to where this line intersects the square d.

Now I have two similar triangles: one whose base goes from the center of the red circle to the center of the lower left circle, and one whose base goes from the center of the red circle to the center of the lower right circle. Calling the angle of my line from the horizontal theta, I get:

tan(theta)=d/s=(s-d)/(2s)

-> 2d=s-d -> d=(1/3)s

-> tan(theta)=1/3 -> theta=arctan(1/3)

[Guest]

I flipped the image and placed the images over top so the 4 black circles line up. Doing this means that the red circle is in the same location on the opersite side. I drew a line through the centre of both red circles sepetating the image in half. Solving Graphically instead of mathmatically (for all the people who dont know arccsine)

Edit:explaination

Edited October 5, 2008 by taliesin

[Guest]

Edit:explaination

I flipped the image and placed the images over top so the 4 black circles line up. Doing this means that the red circle is in the same location on the opersite side. I drew a line through the centre of both red circles sepetating the image in half. Solving Graphically instead of mathmatically (for all the people who dont know arccsine)

How simply explained my friend Well done !!!

[Guest]

Any line through the red dot will divide this 5th circle into half so this one is no problem. for the remaining 4 circles the top left and bottom right will balance each other's area. Now we can adjust the angle of the line so that it passes through bottom left and top right of the 4 circles easily such that the area above the line in bottom left circle is equal to are below the line in top right.

[Guest]

We can assume another circle of the same size of others at the right side of top right circle. Then a line through the red dot and the centre of that assumed circle should do the trick. Naturally we will assume it to be at the same distance from the top right as this circle with red dot is from its neighbour.

[Guest]

Edit:explaination

I flipped the image and placed the images over top so the 4 black circles line up. Doing this means that the red circle is in the same location on the opersite side. I drew a line through the centre of both red circles sepetating the image in half. Solving Graphically instead of mathmatically (for all the people who dont know arccsine)

Kudos to you! taliesin

[Guest]

How simply explained my friend Well done !!!

Kudos to you! taliesin

Thanks guys. Was reading the other (mathmatical answers) and was starting to think this was super hard, until the simple solutionhit me

[Yoruichi-san]

Thanks guys. Was reading the other (mathmatical answers) and was starting to think this was super hard, until the simple solutionhit me

Lol...there was only one answer...that was me...and I thought it was simple...;P

Reminds me of a story I heard about this genius scientist (way smarter than I could ever be)...

One day some one asked him the bird flying b/w two trains problem, i.e. there are two trains coming towards each other and a bird that flies back and forth between the two trains. The question is what distance does the bird fly before the trains collide.

So this genius scientist thinks about it for a little while and then gives the answer. The guy who asked the question was like "Wow, you figured out really quickly that all you have to do is find the total time the trains are moving and multiply by the speed of the bird. Most people don't realize that, they try to construct an infinite geometric series and find the sum..."

And the genius scientist gives the guy a look and is like "but that's exactly what I did!"

True story, I heard it from a professor at college.

Edit: Added smiley ;P

Edited October 6, 2008 by Yoruichi-san