## Friday, July 25, 2008

### Triangle Trivia

You have a normal paper in which a triangle is inscribed.

For a circular piece of paper, the lines along which you can fold it "in half" (with half the area on each side) are precisely the lines through the center. Triangles are different and folding it exactly in half won't be that easy, even for an expert geometrician like you, would it ?

Do let me know how you do it, and don't assume anything !

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Surge said...

Area of a triangle is one-half of length of a side times its height, which the distance between the side and the opposite corner (along the perpendicular, of course).

An aside: this means if you have two parallel lines, any triangle with one side as a given segment on one line, and the opposite corner on the other line, will have the same area, regardless of where you slide that corner on its line.

Back to the problem:
The area formula means if we can fold the paper so, that the fold line starts at a corner and divides the opposite side in half, the parts on either side of the fold will have equal areas. This is because the height for either half will be the same, since it is the distance from the same line to the same point (corner).

Fold the paper, so that the fold crosses corner A, and corners B and C match.

July 26, 2008 12:21 PM
Anonymous said...

aww you beat me to it, and i just got home,

~ghostrider655

July 26, 2008 12:59 PM
Euclid's Brother said...

Surge, unless it's an isocelles triangle, corners won't match..

Your are right that the fold should start in one corner and end at the midpoint of the opposite side. The two sides won't match each other in shape, but they will have the same area.

July 26, 2008 5:19 PM
Surge said...

Euclid's brother is absolutely correct - nothing like a smart bother to teach you to think things through, I guess :)

My theory was good, but the execution sucked...

First fold to match up the corners on a side - this gives you the side's midpoint. Then make the real fold from the opposite corner to the midpoint found.

Just did the folding with a real piece of paper - looks really spiffy :)

July 26, 2008 7:18 PM