考题22 2015 AMC 10B
Problem 22In the figure shown below, is a regular pentagon and . What is ?
Solution 1Triangle is isosceles, so . since is also isosceles. Using the symmetry of pentagon , notice that . Therefore, .
So, since must be greater than 0.
Notice that .
Solution 2 (Trigonometry)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 not permitted in the 2015 AMC 10B, we can not use a calculator to find out which of the options is equal to , but we can find that this is closest to .
Solution 3When you first see this problem you can´t help but see similar triangles. But this shape is filled with triangles throwing us off. First, let us write our answer in terms of one side length. I chose to write it in terms of so we can apply similar triangles easily. To simplify the process lets write as .
First what is in terms of , also remember :
Next, find in terms of , also remember :
So adding all the we get . Now we have to find out what x is. For this, we break out a bit of trig. Let´s look at By the law of sines:
Now by the double angle identities in trig. substituting in
A good thing to memorize for AMC and AIME is the exact values for all the nice sines and cosines. You would then know that: =
so now we know:
Substituting back into we get