Difference between revisions of "2021 AMC 12B Problems/Problem 15"
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==Video Solution by Hawk Math== | ==Video Solution by Hawk Math== | ||
https://www.youtube.com/watch?v=p4iCAZRUESs | https://www.youtube.com/watch?v=p4iCAZRUESs | ||
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+ | ==Video Solution by TheBeautyofMath== | ||
+ | https://youtu.be/FV9AnyERgJQ?t=1226 | ||
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+ | ~IceMatrix | ||
==See Also== | ==See Also== |
Revision as of 06:51, 19 February 2021
Contents
Problem
The figure is constructed from line segments, each of which has length
. The area of pentagon
can be written is
, where
and
are positive integers. What is
Solution 1
Let be the midpoint of
. Noting that
and
are
triangles because of the equilateral triangles,
. Also,
and so
.
Solution 2
Draw diagonals and
to split the pentagon into three parts. We can compute the area for each triangle and sum them up at the end. For triangles
and
, they each have area
. For triangle
, we can see that
and
. Using Pythagorean Theorem, the altitude for this triangle is
, so the area is
. Adding each part up, we get
.
Video Solution by OmegaLearn (Extending Lines, Angle Chasing, Trig Area)
~ pi_is_3.14
Video Solution by Hawk Math
https://www.youtube.com/watch?v=p4iCAZRUESs
Video Solution by TheBeautyofMath
https://youtu.be/FV9AnyERgJQ?t=1226
~IceMatrix
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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 |
2021 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.