1994 AHSME Problems/Problem 23
Contents
Problem
In the -plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at and . The slope of the line through the origin that divides the area of this region exactly in half is
Solution
Let the vertices be . It is easy to see that the line must pass through . Let the line intersect at the point (i.e. the point units below ). Since the quadrilateral and pentagon must have the same area, we have the equation . This simplifies into , or , so . Therefore the slope of the line is
Solution 2
Consider the small rectangle between and with area . If we exclude that area then the shape becomes a much simpler, its just a rectangle. In order to offset the exclusion of that area from the shape below the line, we must also exclude a shape with the same area above the line. We want the remainder after excluding both areas to be simple, so exclude a rectangle between and and and for some unknown . For the area to be the same we need , or . After excluding our two offsetting areas, we're left with a rectangle from to and to . The area of this region is clearly bisected by its diagonal line. The line passes through and so its slope is and the answer is
See Also
1994 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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