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 Pythagoras, 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
. Solving this, we 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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