An unit square is divided into four regions by a diagonal and a line that connects one of the remaining vertices to the midpoint of one of the opposite sides.
What are the areas of the four regions?
Source: Mathematician Ed Barbeau (University of Toronto 1995).
Directions on the below description assume diagonal is from upper left to lower right and the starting vertex of the second line is at the lower left and goes to the midpoint of the side on the right.
The point of intersection of the two internal lines, (x,y) can be found from:
x = 2y and
y = 1  x
y = 1  2y
3y = 1
y = 1/3
x = 2/3
The bottom triangle has height 1/3 and base 1, for an area of 1/6.
The left side triangle has height 2/3 and base 1, for an area of 1/3.
The right side triangle has height 1/3 and base 1/2 for an area of 1/12.
The remaining quadrilateral then has area 1  1/6  1/3  1/12 = 1  7/12 = 5/12.

Posted by Charlie
on 20160517 12:45:44 