Ageometry student wants to draw a rectangle inscribed in a semicircle of radius of 8. if one side must be on the semicircle's diameter, what is the area of the largest rectangle that the student can draw?

For the answer to the question above, let's start with the whole circle.
Let's assume that the maximum possible area of a rectangle inscribed in a complete circle is achieved when the rectangle is a square. 

D= Circle's Diameter = 16 
square's area = (D^2) / 2 = 256/2 =128 
Imagine we want to break the circle into two semicircles, the square would be divided into two rectangles which would have the maximum possible area. 
rectangle's area = square's area / 2 = 128/2 = 64

To let it have a splended look

Hey here is your answer