Difference between revisions of "2014 AMC 10A Problems/Problem 13"
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==Solution 2== | ==Solution 2== | ||
− | As seen in the previous solution, segment <math>GH</math> is <math>\sqrt{3}</math>. Think of the picture as one large equilateral triangle, <math>\triangle{JKL} with the sides of < | + | As seen in the previous solution, segment <math>GH</math> is <math>\sqrt{3}</math>. Think of the picture as one large equilateral triangle, <math>\triangle{JKL}</math> with the sides of <math>2\sqrt{3}+1</math>, by extending <math>EF</math>, <math>GH</math>, and <math>DI</math> to points <math>J</math>, <math>K</math>, and <math>L</math>, respectively. This makes the area of <math>\triangle{JKL}</math> <math>\dfrac{\sqrt{3}}{4}(2\sqrt{3}+1)^2=\dfrac{12+13\sqrt{3}}{4}</math>. |
<asy> | <asy> |
Revision as of 13:04, 13 February 2014
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
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
, respectively. This makes the area of
.
Triangles ,
, and
have sides of
, so their total area is
.
Now, you subtract their total area from the area of :
(Solution by Pyson)
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 |
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