First, make a sketch . . .
[I'll use Calculus to maximize OC.]
A semi-circle is dawn outwardly on a chord of a circle with center and raidus 1.
The point on this semi-circle is the furthest from center
Of course, the length of is dependant on the choice of chord
Determine so that is maximum.Code:* * * * * A * o * / :* o 1/ : o * / θ D: * * O * - - - + * - - o C * \ : * 1\ : o * \ :* o * o * * B * * *
The circle with center has radii
A semicircle is constructed (outward) with center and diameter
. . It has radii:
In right triangle
To maximize , set
We have: .
. . Hence: .
is maximum when
Then and is an isosceles right triangle.
Since its legs are length 1, its hypotenuse is: .