bonanova Posted February 28, 2017 Report Share Posted February 28, 2017 A triangle has sides of length {1, 1, sqrt(2)}. What is the length of the shortest cut that divides the triangle into two pieces that have equal areas? Quote Link to comment Share on other sites More sharing options...

0 Pickett Posted February 28, 2017 Report Share Posted February 28, 2017 (edited) I'll take a quick stab at it: Spoiler ^{sqrt(2π)} /_{ 4 }= ~0.62666 The "obvious" first answer is ^{sqrt(2)} / _{2} (bisect the 90 degree angle perpendicular to the hypotenuse)...But if we step back and look at it a different way: The current triangle has an area of ^{1}/_{2}... So, we simply need to make a cut that leaves two areas of ^{1}/_{4} (not necessarily similar shapes). The shortest STRAIGHT cut would be the ^{sqrt(2)} / _{2}, but if we decide to make a curved cut, we can get it shorter. I'll try to describe it (can't add pictures right now) Choose one of the 45 degree angle points and center a circle there with radius r. Mark the points along the hypotenuse and adjacent leg at points to create a segment of a circle with a 45 degree angle and radius r. We know, then, that is exactly ^{1}/_{8} of the area of the full circle. Since we are trying to get an area of ^{1}/_{4}, and we have ^{1}/8 of a circle, the full area of the circle would be 2. so we can figure out the length of r by doing πr^{2}=2. Leaving us with r = ^{sqrt(}^{2π}^{) }/ _{π} Given that, we need to find the length of just that arc of the circle...which is ^{1}/_{8} of the perimeter. So we know P=πd...which leaves us with P=2*sqrt(2π). Divide it by 8, and you get ^{sqrt(2π)} /_{ 4} as the length of the arc connecting your points...leaving 2 pieces of the triangle with areas of ^{1}/_{4} each. Edited February 28, 2017 by Pickett It duplicated my response for some reason.... Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted March 1, 2017 Author Report Share Posted March 1, 2017 Nice stab! I made a picture of it. Spoiler Quote Link to comment Share on other sites More sharing options...

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A triangle has sides of length {1, 1, sqrt(2)}.

What is the length of the shortest cut that divides the triangle into two pieces that have equal areas?

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