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Difference between revisions of "2006 Alabama ARML TST Problems/Problem 7"

(New page: ==Problem== Four equilateral triangles are drawn such that each one shares a different side with a square of side length 10. None of the areas of the triangles overlap with the area of the...)
 
(→‎Solution 2)
 
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</asy></center>
 
</asy></center>
  
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Since all the angles in the equilateral triangles are <math>60^\circ</math>, all the angles in the square are <math>90^\circ</math>, the angle between the sides of two adjacent equilateral triangles is <math>360^\circ - (90^\circ + 2\cdot60^\circ) = 150^\circ</math>.  
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Since all the angles in the equilateral triangles are <math>60^\circ</math>, all the angles in the square are <math>90^\circ</math>, the anssen the sides of two adjacent equilateral triangles is <math>360^\circ - (90^\circ + 2\cdot60^\circ) = 150^\circ</math>.  
  
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By the [[Law of Cosines]], the square of the length of the side of the larger square, which is also the area of the larger square, is <math>x^2 = 10^2 + 10^2 - 2\cdot10\cdot10\cdot \cos{150^\circ} = 200+100\sqrt{3}</math>.  
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By the [[Law of Cosines]], the square of the length of the side of the larger square, which is also the area of the larger square, is <math>x^2 = 10^2 + 10^2 - 2\cdot10\cdot10\cdot \cos{150^\circ} = 200+100\sqrt{3}</math>.
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==Solution 2==
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Since we know that a square is a rhombus, and a rhombus's area can be calculated by the product of its diagonals divided by two, we simply find the length of the diagonals, which is
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<math>10\sqrt{3}+10</math> each, and find the area, which is
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<math>\frac{(10\sqrt{3}+10)^2}{2}=\boxed{200+100\sqrt{3}}</math>
  
 
==See Also==
 
==See Also==
  
 
{{ARML box|year=2006|state=Alabama|num-b=6|num-a=8}}
 
{{ARML box|year=2006|state=Alabama|num-b=6|num-a=8}}

Latest revision as of 22:47, 2 September 2024

Problem

Four equilateral triangles are drawn such that each one shares a different side with a square of side length 10. None of the areas of the triangles overlap with the area of the square. The four vertices of the triangles that aren’t vertices of the square are connected to form a larger square. Find the area of this larger square.

Solution

[asy] defaultpen(linewidth(0.7)); draw((13.66,0)--(5,5)--(0,13.66)--(-5,5)--(-13.66,0)--(-5,-5)--(0,-13.66)--(5,-5)--cycle); draw((5,5)--(-5,5)--(-5,-5)--(5,-5)--cycle); draw((13.66,0)--(0,13.66)--(-13.66,0)--(0,-13.66)--cycle); [/asy]

Since all the angles in the equilateral triangles are $60^\circ$, all the angles in the square are $90^\circ$, the anssen the sides of two adjacent equilateral triangles is $360^\circ - (90^\circ + 2\cdot60^\circ) = 150^\circ$.

By the Law of Cosines, the square of the length of the side of the larger square, which is also the area of the larger square, is $x^2 = 10^2 + 10^2 - 2\cdot10\cdot10\cdot \cos{150^\circ} = 200+100\sqrt{3}$.

Solution 2

Since we know that a square is a rhombus, and a rhombus's area can be calculated by the product of its diagonals divided by two, we simply find the length of the diagonals, which is

$10\sqrt{3}+10$ each, and find the area, which is

$\frac{(10\sqrt{3}+10)^2}{2}=\boxed{200+100\sqrt{3}}$

See Also

2006 Alabama ARML TST (Problems)
Preceded by:
Problem 6 		Followed by:
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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