1996 AHSME Problems/Problem 20
Problem 20
In the xy-plane, what is the length of the shortest path from to that does not go inside the circle ?
Solution
The pathway from to will consist of three segments:
1) , where is tangent to the circle at point .
2) , where is tangent to the circle at point .
3) , where is an arc around the circle.
The actual path will go , so the acutal segments will be in order .
Let be the center of the circle at .
and since is on the circle. Since $\trianlge OAB$ (Error compiling LaTeX. ! Undefined control sequence.) is a right triangle with right angle , we find that . This means that is a triangle with sides .
Notice that is a line, since all points are on . In fact, it is a line that makes a angle with the positive x-axis. Thus, , and . These are two parts of the stright line . The third angle is , which must be as well. Thus, the arc that we travel is a arc, and we travel around the circle.
Thus, , , and . The total distance is , which is option .
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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