2015 AMC 10B Problems/Problem 22
In the figure shown below, is a regular pentagon and . What is ?
Triangle is isosceles, so . since is also isosceles. Using the symmetry of pentagon , notice that . Therefore, .
So, since must be greater than 0.
Notice that .
Note by Fasolinka: Alternatively, extend and call its intersection with . It is not hard to see that quadrilaterals and are parallelograms, so , and the same result is achieved.
Notice that . Since a triangle has the congruent sides equal to times the short base side, we have . Now notice that , and that is . So, and adding gives , or .
When 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
Notice that and , so we have . Thus Solving the equation gets .
Since Solving the equation gets .
Since Solving the equation gets
Finally adding them up gets
Note: this solution might be a bit complicated but it definitely works when none of the cleverer symmetries in Solution 1 is noticed.
Note that: Summing the equations, we have: We know that So we have All that remains is to find By the law of cosines, we have
I'll leave you to convince yourself about certain facts that were deduced in order to obtain these equations.
https://www.youtube.com/watch?v=TpHZVbBGmVQ (Beauty of Math)
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