2015 AMC 10B Problems/Problem 22
Contents
Problem
In the figure shown below, is a regular pentagon and . What is ?
Solution 1
Triangle is isosceles, so . since is also isosceles. Using the symmetry of pentagon , notice that . Therefore, .
Since , .
However, since must be greater than 0.
Notice that .
Therefore,
Solution 2
Note that since is a regular pentagon, all of its interior angles are . We can say that pentagon is also regular by symmetry. So, all of the interior angles of are . Now, we can angle chase and use trigonometry to get that , , and . Adding these together, we get that . Because calculators were permitted in the 2015 AMC 10B, we can use a calculator to find out which of the options is equal to . We can find that the option that matches is $\boxed{\mathbf{(D)}\ 1+\sqrt{5}$.
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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All AMC 10 Problems and Solutions |
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