Geometry Puzzle

[Thomas154321]

Draw a right-angled triangle.

3 ropes are attached to the midpoints of each side, and each rope is half the length of the side it is attached to. A goat is tied to the end of the rope on the hypotenuse, and two sheep are attached to the ropes on the shorter sides. The area of the triangle is 1 acre. Find the area that the goat can reach that the sheep cannot.


~
I'm not sure how hard this should be - it was written by a university professor for my Uncle who has limited mathematical knowledge but enjoys puzzles. I can't think where to start with it though. Thanks for any help.

[Rassis]

If I understood it correctly, it seems that the sheep can reach everywhere inside the triangle or does its exterior count as well?

[Thomas154321]

I've drawn a diagram of it. We want the red area. The blue circles represent the outside of where the sheep can reach.

[Rassis]

First consider a generic circle with radius R and a rope connecting any two points. The area A limited by the rope and the circle is given by:

A = R^2*[a/2 – sin(a/2)*cos(a/2)]

Where “a” is the angle formed by the two lines that connect the center of the circle and each of the two extreme points of the rope. Let us call it the inner-angle (perhaps the expression is not the most correct in English…).

Second, consider a right-angled triangle with sides X and Y and hypotenuse Z.



The inner-angle “a1” of the circle with diameter Y is given by:

a1 = (Pi) – 2*arc.tan(X/Y)

And the inner-angle “a2” of the circle with diameter X is given by:

a2 = (Pi) – 2*arc.tan(Y/X)

If the area of the triangle is S, then S = ½*(X*Y)

Now, suppose X = 1. Because it is assumed that the area of the triangle S = 1, you have: Y = 2*1/1 = 2. And you get sequentially:

a1 = 2.2142974
a2 = 0.9272952
Ax = 0.0159119
Ay = 0.7071487

And finally the area A we are after is:

A = Az – Ax – Ay = (Pi)/8*(X^2 + Y^2) – Ax – Ay = 1.9634954 – 0.0159119 – 0.7071487 = 1.2404348

If X = Y = 1.4142136, you get A = 1.2853982 which is a maximum.

Rui

[Thomas154321]

Super! I'll send it to my uncle and see if that's what the answer was supposed to be. That all makes sense though. Thanks.

[Rassis]

It was a pleasure.

Rui