Difference between revisions of "2012 AMC 12B Problems/Problem 21"
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And since <math>k = \frac{\sin(120^{\circ})}{x}</math>, then: | And since <math>k = \frac{\sin(120^{\circ})}{x}</math>, then: | ||
− | <cmath>x = \frac{\sqrt{3}}{2}\ | + | <cmath>x = \frac{\sqrt{3}}{2}\cdot58</cmath> |
<cmath>x = \boxed{29\sqrt{3}}</cmath> | <cmath>x = \boxed{29\sqrt{3}}</cmath> | ||
Revision as of 02:26, 19 January 2021
Problem 21
Square is inscribed in equiangular hexagon
with
on
,
on
, and
on
. Suppose that
, and
. What is the side-length of the square?
(diagram by djmathman)
Solution
Extend and
so that they meet at
. Then
, so
and therefore
is parallel to
. Also, since
is parallel and equal to
, we get
, hence
is congruent to
. We now get
.
Let ,
, and
.
Drop a perpendicular line from to the line of
that meets line
at
, and a perpendicular line from
to the line of
that meets
at
, then
is congruent to
since
is complementary to
. Then we have the following equations:
The sum of these two yields that
So, we can now use the law of cosines in :
Therefore
Solution 2
First, we want to angle chase. Set equal to
degrees.
Now the key idea is that you want to relate the numbers that you have. You know and that
. We proceed with the Law of Sines.
Call the side length of the square x. Then we are going to set a constant k equal to , and this is consistent for every triangle in the diagram because all the angles of the hexagon are equiangular (and so they are all
).
Then we get the following process:
And now expanding using our trig formulas, we get:
And so now we have a triangle where and
. Put them in a triangle where the hypotenuse is 1. Then, by the Pythagorean Theorem, we get:
And since , then:
Solution by IronicNinja
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2012amc12b/273
~dolphin7
See Also
2012 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.