2018 AMC 10B Problems/Problem 17
Problem
In rectangle , and . Points and lie on , points and lie on , points and lie on , and points and lie on so that and the convex octagon is equilateral. The length of a side of this octagon can be expressed in the form , where , , and are integers and is not divisible by the square of any prime. What is ?
Solution 1
Let . Then .
Now notice that since we have .
Thus by the Pythagorean Theorem we have which becomes .
Our answer is . (Mudkipswims42)
Solution 2
Denote the length of the equilateral octagon as . The length of can be expressed as . By the Pythagorean Theorem, we find that: Since , we can say that . We can discard the negative solution, so ~ blitzkrieg21
Solution 3
Let the octagon's side length be . Then and . By the Pythagorean theorem, , so . By expanding the left side and combining the like terms, we get . Solving this using the quadratic formula, , we use , , and , to get one positive solution, , so
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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