2015 AMC 8 Problems/Problem 2
Point is the center of the regular octagon , and is the midpoint of the side What fraction of the area of the octagon is shaded?
Since octagon is a regular octagon, it is split into equal parts, such as triangles , etc. These parts, since they are all equal, are of the octagon each. The shaded region consists of of these equal parts plus half of another, so the fraction of the octagon that is shaded is
The octagon has been divided up into identical triangles (and thus they each have equal area). Since the shaded region occupies out of the total triangles, the answer is .
For starters, what I find helpful is to divide the whole octagon up into triangles as shown here:
Now, it is just a matter of counting the larger triangles. Remember that and are not full triangles and are only half for these purposes. We count it up and we get a total of of the shape shaded. We then simplify it to get our answer of .
We can divide the octagon into 8 parts and pretend that the area is 64. We know that X is the midpoint of BA and that each space between two points is 8 because 64/8=8. This means that BX=4 because 8/2=4. Then, we add that to 3*8 because there are 3 spaces between points that are each 8. After that, you turn it into a fraction, 28/64, and simplify to get .
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