Difference between revisions of "2017 AMC 8 Problems/Problem 25"
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~savannahsolver | ~savannahsolver | ||
Latest revision as of 11:06, 24 July 2024
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?
![[asy]draw((1,1.732)--(2,3.464)--(3,1.732)); draw(arc((0,0),(2,0),(1,1.732))); draw(arc((4,0),(3,1.732),(2,0))); label("$U$", (2,3.464), N); label("$S$", (1,1.732), W); label("$T$", (3,1.732), E); label("$R$", (2,0), S);[/asy]](http://latex.artofproblemsolving.com/7/9/f/79f3ffcfa12364f64959c7aeadc809ba8d2aa6ac.png)
Solution 2
In addition to the given diagram, we can draw lines 
and 
The area of rhombus 
is half the product of its diagonals, which is 
. However, we have to subtract off the circular segments. The area of those can be found by computing the area of the circle with radius 2, multiplying it by 
, then finally subtracting the area of an equilateral triangle with a side length 2 from the sector. The sum of the areas of the circular segments is 
The area of rhombus minus the circular segments is 
~PEKKA
Video Solutions
~savannahsolver
