1989 AIME Problems/Problem 6
Two skaters, Allie and Billie, are at points and , respectively, on a flat, frozen lake. The distance between and is meters. Allie leaves and skates at a speed of meters per second on a straight line that makes a angle with . At the same time Allie leaves , Billie leaves at a speed of meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
Label the point of intersection as . Since , and . According to the law of cosines,
Since we are looking for the earliest possible intersection, seconds are needed. Thus, meters is the solution.
Alternatively, we can drop an altitude from and arrive at the same answer.
Let be the point of intersection between the skaters, Allie and Billie. We can draw a line that goes through and is parallel to . Letting this line be the -axis, we can reflect over the -axis to get . As reflections preserve length, .
We then draw lines and . We can let the foot of the perpendicular from to be , and we can let the foot of the perpendicular from to be . In doing so, we have constructed rectangle .
By , we have and , where is the number of seconds it takes the skaters to meet. Furthermore, we have triangle , so , and . Since we have , . By Pythagoras, .
As is a rectangle, . Thus , so we get . Solving for , we find .
Our answer, , is equivalent to . Thus, .
We can define to be the time elapsed since both Allie and Billie moved away from points and respectfully. Also, set the point of intersection to be . Then we can produce the following diagram:
Now, if we drop an altitude from point, we get :
We know this from the triangle that is formed. From this we get that:
Therefore, we get that or . Since , we have that (since the problem asks for the quickest possible meeting point), so the distance Allie travels before meeting Billie would be meters.
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