2017 AMC 8 Problems/Problem 25
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 1
label("", Y, E);
</asy>
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
- Happytwin
~savannahsolver
Solution 3 & Video
First, we must make it like an equilateral triangle. If you add two of the same arcs, you will get an equilateral triangle. The goal is to find ONE of the arcs and multiply it by two. Then subtract it from the equilateral triangle. If you look at the triangle, T and S are midpoints of the equilateral triangle. So now we can find the arc.These arcs are so now you multiply by 2 so
. And subtract. Equilateral's area is
. So now you do
. That is the answer and thank you.
~ math.is.hard.to.understand
Here's a video to understand it (credits to ~ pi_is_3.14)
https://youtu.be/j3QSD5eDpzU?t=1350
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
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