Final answer:
The slope of the line passing through the origin and bisecting the parallelogram into two congruent polygons is 199/38. To find this, calculate the midpoint of the parallelogram's diagonals and form a slope ratio with the origin. The sum of the numerator and denominator after simplifying the slope is 237.
Step-by-step explanation:
To determine the slope of a line that bisects a parallelogram into two congruent polygons through the origin, we need to locate the midpoint of the parallelogram's opposite sides and then calculate the slope of the line connecting this midpoint to the origin. The midpoints of the opposite sides can be found by averaging the x-coordinates and y-coordinates of the vertices. Once we have the midpoint, the slope (m) is the ratio of the rise (change in y) to the run (change in x), when connecting the origin to the midpoint.
Given the parallelogram vertices (10,45), (10,114), (28,153), and (28,84), the midpoints are (19,99.5) and (19,64.5). However, a line that bisects the parallelogram into two congruent shapes must pass through the midpoint of the diagonals, which when calculated gives us the point (19, 99.5). Starting from the origin (0,0) and passing through this midpoint, the slope would simply be 99.5/19, which in simplest form is 995/190. Both numbers share a common factor of 5, so the fraction simplifies to 199/38, which cannot be simplified further as 199 and 38 are relatively prime. Therefore, the slope of the line is 199/38, and the sum of m and n, i.e., 199 + 38, equals 237.