Half the Area of a Square Field

Suppose a goat is tethered inside a square field with the tether fastened to a point on the side of the field. Relative to the length of the side of the field, what would be the length of the tether so that the goat can graze over HALF of the area of the field?

In the first case, consider the tether fastened to a corner of the field. Then if the goat can graze over half of the field, we can write

From this we can solve for r in terms of s or for the ratio of r to s.

Next consider the case where the tether is fastened to the center of one side.

It seems clear that r in this case will be less than when the tether was fastened to a corner. Why?

Solve for r in terms of s.

Next, consider the case where the tether is fastened to a point on a side NOT at a corner or in the middle.

One such case is on the right.

Again, solve for r in terms of s.

One special case is when the tether is attached to a point on the side, NOT at a corner or the middle, and the length of the tether reaches the adjacent corner of the field.

What is the size or r, relative to s, in that case.

Can you express r as a function of the distance x from a vertex as x goes from 0 to s/2?

Describe the graph of that function.