2008 AMC 12A Problems/Problem 22
- The following problem is from both the 2008 AMC 12A #22 and 2008 AMC 10A #25, so both problems redirect to this page.
A round table has radius . Six rectangular place mats are placed on the table. Each place mat has width and length as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length . Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is ?
Solution 1 (trigonometry)
Let one of the mats be , and the center be as shown:
Since there are mats, is equilateral. So, . Also, .
By the Law of Cosines: .
Since must be positive, .
Solution 2 (without trigonometry)
Draw and as in the diagram. Draw the altitude from to and call the intersection
As proved in the first solution, . That makes a triangle, so and
Since is a right triangle,
Solving for gives
Looking at the diagram above, we know that is a diameter of circle due to symmetry. Due to Thales' theorem, triangle is a right triangle with . lies on and because is also a right angle. To find the length of , notice that if we draw a line from to , the midpoint of line , it creates two - - triangles. Therefore, .
Use the Pythagorean theorem on triangle , we get Using the pythagorean theorem to solve, we get must be positive, therefore