2017 AIME II Problems/Problem 10
Problem
Rectangle has side lengths and . Point is the midpoint of , point is the trisection point of closer to , and point is the intersection of and . Point lies on the quadrilateral , and bisects the area of . Find the area of .
Solution
Impose a coordinate system on the diagram where point is the origin. Therefore , , , and . Because is a midpoint and is a trisection point, and . The equation for line is and the equation for line is , so their intersection, point , is . Using the shoelace formula on quadrilateral , or or drawing diagonal and using , we find that its area is . Therefore the area of triangle is =1092. Using A=, we get 1092= = 42*h. Simplifying, we get h=52. This means that the x-coordinate of P= 84-52=32. Since P lies on , you can solve and get that the y-coordinate of P is 13. Therefore the area of is .
Solution Altered By conantwiz2023
and the distance from to line is and its -coordinate is . Because lies on the equation , 's -coordinate is , which is also the height of . Therefore the area of is .
See Also
2017 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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