Difference between revisions of "2017 AMC 8 Problems/Problem 25"
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==Video Solutions== | ==Video Solutions== | ||
+ | https://youtu.be/LT4gyH--328 | ||
https://youtu.be/wc5rGulTTR8 - Happytwin | https://youtu.be/wc5rGulTTR8 - Happytwin |
Revision as of 20:48, 14 June 2021
Contents
Problem
In the figure shown, and
are line segments each of length 2, and
. Arcs
and
are each one-sixth of a circle with radius 2. What is the area of the region shown?
Solution
Let the centers of the circles containing arcs and
be
and
, respectively. Extend
and
to
and
, and connect point
with point
.
We can clearly see that
is an equilateral triangle, because the problem states that
. We can figure out that
and
because they are
of a circle. The area of the figure is equal to
minus the combined area of the
sectors of the circles(in red). Using the area formula for an equilateral triangle,
where
is the side length of the equilateral triangle,
is
The combined area of the
sectors is
, which is
Thus, our final answer is
Video Solutions
https://youtu.be/wc5rGulTTR8 - Happytwin
https://youtu.be/uvwLT5xBNdU ~DSA_Catachu
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.