2015 AIME I Problems/Problem 6
Problem
Point and are equally spaced on a minor arc of a circle. Points and are equally spaced on a minor arc of a second circle with center as shown in the figure below. The angle exceeds by . Find the degree measure of .
Solution 1
Let be the center of the circle with on it.
Let be the degree measurement of in circle
and be the degree measurement of in circle .
is, therefore, by way of circle and by way of circle . is by way of circle , and by way of circle .
This means that:
which when simplified yields or Since: and So: is equal to + , which equates to . Plugging in yields , or .
Solution 2
Let be the degree measurement of . Since lie on a circle with center , .
Since , . Adding and gives , and . Since is parallel to , .
We are given that are evenly distributed on a circle. Hence,
Here comes the key: Draw a line through parallel to , and select a point to the right of point .
= + = .
Let the midpoint of be , then . Solving gives
The rest of the solution proceeds as in solution 1, which gives
Solution 3
Let
Let arc
Therefore
See Also
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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