Can someone confirm if my reasoning is correct here:

if we make a line from B to C, the triangle on the left (EBD) will equal the triangle on the right (BDC). The left triangle has a hypotenuse of r, base of r/2 and since both triangles will share the height and base, the length of line BC will be r.

The area of the shaded region will be bigger as it includes one triangle plus the remaining shaded area?