Answer to Question #273073 in Calculus for Tomi

Question

Let D be a convex region in R2 and let L be a line segment of length I that connects points on the boundary of D. As we move one end of L around the boundary, the other end will also move about this boundary, and the midpoint of L will trace out a curve within D that bounds a (smaller) region R. Using the corollary to Green’s Theorem for finding area, find an expression that relates the area of R to the area of D in terms of the length I of the line segment. [You might start with some simple regions, but you must show this generally.] 

Answer 1

If Ω is a circle of radius R, then Γ is again a circle with radius r<R

that can be easily computed.

If Ω is a rectangle, then Γ is the same rectangle with a quarter of circle taken out in each corner, see he picture below. Again computations can be made exactly.

in both cases

$Area( Γ)=Area(Ω)-\frac{\pi}{4}l^2$

That is, the area of Γ is the area of Ω minus the area of a circle of radius l/2

hence the final expression will be

$Area( Γ)=Area(Ω)-\frac{\pi}{4}l^2$