Question

The capital letter F can be formed by connecting the points (0,4),(5,4),(5,2),(2,2),(2,0),(4,0),(4,-2),(2,-2),(2,-7), and (0,-7), in this order. Determine the slope of the line that passes through the origin and divides the letter F into two regions of equal areas.

Answer 1

Draw the figure. The area is 32.
The area above the x-axis is 14, and below is 18.

So, we want a line sloping down through the origin.

Further inspection shows that the area of the foot is 10, so we want a chunk of area 6 in the middle section.

That will be formed by a trapezoid of height 4 with bases 2 and b, such that

(2+b)/2 * 4 = 6
2+b = 3
b = 1

So, we want the line through (0,0) and (4,-1). That will have slope -1/4.