2014 AMC 10A Problems/Problem 13
Contents
Problem
Equilateral has side length , and squares , , lie outside the triangle. What is the area of hexagon ?
Solution 1
The area of the equilateral triangle is . The area of the three squares is .
Since , .
Dropping an altitude from to allows to create a triangle since is isosceles. This means that the height of is and half the length of is . Therefore, the area of each isosceles triangle is . Multiplying by yields for all three isosceles triangles.
Therefore, the total area is .
Solution 2
I have no idea how to draw pictures with LaTex, so someone please help me with this:
As seen in the previous solution, segment is . Think of the picture as one large equilateral triangle, with the sides of , by extending , , and to points , , and . This makes the area of .
The extra equilateral triangles have sides of , so their areas are .
Now, you subtract the areas of the 3 equilateral triangles:
(Solution by Pyson)
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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All AMC 10 Problems and Solutions |
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