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?
Contents
Solution 1
Since octagon is a regular octagon, it is split into 8 equal parts, such as triangles , etc. These parts, since they are all equal, are of the octagon each. The shaded region consists of 3 of these equal parts plus half of another, so the fraction of the octagon that is shaded is
Solution 2
The octagon has been divided up into 16 identical triangles (and thus they each have equal area). Since the shaded region occupies 7 out of the 16 total triangles, the answer is .
Solution 3
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: or .
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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