Let A=(0,0), B=(2,4), C=(17,4) and D=(15,0). Then ABCD is a parallelogram. A line through the point (0,−1) divides the parallelogram into two regions of equal area. The slope of this line can be written as a/b where a and b are positive coprime integers. Find a+b.
math - Steve, Wednesday, June 26, 2013 at 4:05pm
Let the line from (0,-1) intersect the x-axis at (2+h,0)
We know that h>0 because the line with slope 1/2 passes through (10,4) and obviously falls far short of dividing the are into two equal parts.
So, we now have ABCD divided into two triangles of width 2 and height 4, two rectangles of width h and height 4, and the inside rectangle of width 13-2h and height 4.
By symmetry, we need to have
4(2+h) + 2h = 13
h = 5/6
So, the line from (0,-1) through (17/6,0) will do the equal division.
That line has slope 17/6.
math - Steve, Wednesday, June 26, 2013 at 4:20pm
Sorry - the slope is 6/17, but the answer is the same.