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 ,
So, since must be greater than 0.
Notice that .
Therefore,
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.
Solution 2
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 .
Solution 3 (Trignometry)
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
Solution 4
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.
~ Nafer
Solution 5
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
Thus,
I'll leave you to convince yourself about certain facts that were deduced in order to obtain these equations.
~mathboy282
Solution 6(Trignometry)
Let's denote the length FG . From similarity, GH, HI, JI, and JF are also x. Now, because by SAS similarity, we can write the lengths of FG + JH + CD as . We obtain this result by directly writing out the similarity statements and then multiplying.
So all we have to do is find x. Because a regular pentagon has angles of 108 degrees, with an easy angle chase we can find . Now drop the altitude from A, and call the point T. Since degrees, we can easily use sine18 deg to find TG = and therefore x = . Plugging this into , we obtain the result .
~MathCosine
Solution 7 (Knowledge of )
In a pentagon, FG:JH=JH:CD=CD:AB=1:. Also, JH=AG. FG=, so FG+JH+CD=
Video Solution
https://www.youtube.com/watch?v=TpHZVbBGmVQ (Beauty of Math)
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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