# 2015 AMC 10B Problems/Problem 22

Solution

Triangle is isosceles so ==. Using the symmetry of pentagon , notice that is congruent to , so triangles and are congruent to our original triangle . Therefore, = . Now, we still need to find the length of and . Also, we know that = since pentagon is regular. Let's call the length of and . Now we can solve for . Triangles and are similar. So,

From this, we get .

Now, we just have to find the length of which we'll call . We already know that = = . Triangles and are similar so we have,

Solving for we get

Adding up , , and gives us which is

Solution by arebei2