Difference between revisions of "2014 AMC 10A Problems/Problem 13"
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label("$E$",E,W); | label("$E$",E,W); | ||
label("$F$",F,E); | label("$F$",F,E); | ||
− | label("$G$",G, | + | label("$G$",G,SE); |
label("$H$",H,SE); | label("$H$",H,SE); | ||
label("$I$",I,SW); | label("$I$",I,SW); | ||
Line 42: | Line 42: | ||
<math> \textbf{(A)}\ \dfrac{12+3\sqrt3}4\qquad\textbf{(B)}\ \dfrac92\qquad\textbf{(C)}\ 3+\sqrt3\qquad\textbf{(D)}\ \dfrac{6+3\sqrt3}2\qquad\textbf{(E)}\ 6 </math> | <math> \textbf{(A)}\ \dfrac{12+3\sqrt3}4\qquad\textbf{(B)}\ \dfrac92\qquad\textbf{(C)}\ 3+\sqrt3\qquad\textbf{(D)}\ \dfrac{6+3\sqrt3}2\qquad\textbf{(E)}\ 6 </math> | ||
+ | [[Category: Introductory Geometry Problems]] | ||
− | ==Solution== | + | ==Solution 1== |
− | The area of the equilateral triangle is | + | The area of the equilateral triangle is <math>\dfrac{\sqrt{3}}{4}</math>. The area of the three squares is <math>3\times 1=3</math>. |
Since <math>\angle C=360</math>, <math>\angle GCH=360-90-90-60=120</math>. | Since <math>\angle C=360</math>, <math>\angle GCH=360-90-90-60=120</math>. | ||
Line 51: | Line 52: | ||
Therefore, the total area is <math>3+\dfrac{\sqrt{3}}{4}+\dfrac{3\sqrt{3}}{4}=3+\dfrac{4\sqrt{3}}{4}=3+\sqrt{3}\implies\boxed{\textbf{(C)}\ 3+\sqrt3}</math>. | Therefore, the total area is <math>3+\dfrac{\sqrt{3}}{4}+\dfrac{3\sqrt{3}}{4}=3+\dfrac{4\sqrt{3}}{4}=3+\sqrt{3}\implies\boxed{\textbf{(C)}\ 3+\sqrt3}</math>. | ||
+ | |||
+ | ==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}</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> | ||
+ | import graph; | ||
+ | size(10cm); | ||
+ | pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); | ||
+ | pair B = (0,0); | ||
+ | pair C = (1,0); | ||
+ | pair A = rotate(60,B)*C; | ||
+ | |||
+ | pair E = rotate(270,A)*B; | ||
+ | pair D = rotate(270,E)*A; | ||
+ | |||
+ | pair F = rotate(90,A)*C; | ||
+ | pair G = rotate(90,F)*A; | ||
+ | |||
+ | pair I = rotate(270,B)*C; | ||
+ | pair H = rotate(270,I)*B; | ||
+ | |||
+ | pair J = rotate(60,I)*D; | ||
+ | pair K = rotate(60,E)*F; | ||
+ | pair L = rotate(60,G)*H; | ||
+ | |||
+ | draw(A--B--C--cycle); | ||
+ | draw(A--E--D--B); | ||
+ | draw(A--F--G--C); | ||
+ | draw(B--I--H--C); | ||
+ | |||
+ | draw(E--F); | ||
+ | draw(D--I); | ||
+ | draw(I--H); | ||
+ | draw(H--G); | ||
+ | |||
+ | draw(I--J--D); | ||
+ | draw(E--K--F); | ||
+ | draw(G--L--H); | ||
+ | |||
+ | label("$A$",A,N); | ||
+ | label("$B$",B,SW); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,W); | ||
+ | label("$E$",E,W); | ||
+ | label("$F$",F,E); | ||
+ | label("$G$",G,E); | ||
+ | label("$H$",H,SE); | ||
+ | label("$I$",I,SW); | ||
+ | label("$J$",J,SW); | ||
+ | label("$K$",K,N); | ||
+ | label("$L$",L,SE); | ||
+ | </asy> | ||
+ | |||
+ | Triangles <math>\triangle{DIJ}</math>, <math>\triangle{EFK}</math>, and <math>\triangle{GHL}</math> have sides of <math>\sqrt{3}</math>, so their total area is <math>3(\dfrac{\sqrt{3}}{4}(\sqrt{3})^2)=\dfrac{9\sqrt{3}}{4}</math>. | ||
+ | |||
+ | Now, you subtract their total area from the area of <math>\triangle{JKL}</math>: | ||
+ | |||
+ | <math>\dfrac{12+13\sqrt{3}}{4}-\dfrac{9\sqrt{3}}{4}=\dfrac{12+13\sqrt{3}-9\sqrt{3}}{4}=\dfrac{12+4\sqrt{3}}{4}=3+\sqrt{3}\implies\boxed{\textbf{(C)}\ 3+\sqrt3}</math> | ||
+ | |||
+ | ==Solution 3== | ||
+ | We will use, <math>\frac{1}{2}ab\sin x</math> to find the area of the following triangles. Since <math>\angle A=360</math>, <math>\angle EAF=360-90-90-60=120</math>. | ||
+ | |||
+ | The Area of <math>\triangle AEF</math> is <math>\frac{1}{2} \cdot 1 \cdot 1 \cdot \sin(120)</math>. Noting, <math>\sin (2x) = 2\sin (x)\cos (x)</math>, | ||
+ | |||
+ | Area of <math>\triangle AEF = \frac{1}{2} \cdot 1 \cdot 1 \cdot 2 \cdot \sin(60) \cdot \cos(60) = \dfrac{\sqrt{3}}{4}</math>, | ||
+ | |||
+ | Area of <math>\triangle ABC = \frac{1}{2} \cdot 1 \cdot 1 \cdot \sin(60) = \dfrac{\sqrt{3}}{4}</math>, | ||
+ | |||
+ | Area of square ABDE = 1, | ||
+ | |||
+ | Therefore the composite area of the entire figure is, <cmath>3 \cdot [\triangle AEF] + [\triangle ABC] + 3 \cdot [ABDE] = 3 \dfrac{\sqrt{3}}{4} + \dfrac{\sqrt{3}}{4} + 3 \cdot 1 = 4 \dfrac{\sqrt{3}}{4} + 3 = \sqrt{3} + 3 \implies\boxed{\textbf{(C)}\ 3+\sqrt3}</cmath> | ||
+ | |||
+ | |||
+ | |||
+ | ==Solution 4== | ||
+ | We know that the area is equal to 3*EAF+3*ACGF+ABC. We also know that ACGF and the rest of the squares' area is equal to 1. Therefore the answer is 3*EAF+ABC+3. The only one with "+3" or "3+" is C, our answer. Very unreliable. | ||
+ | -Reality Writes | ||
+ | |||
+ | ==Solution 5== | ||
+ | <math>\angle{AEF} = 180- \angle{BAC} = 120</math> | ||
+ | |||
+ | The area of the obtuse triangle is <math>\frac{1}{2}\sin{120} = \frac{\sqrt{3}}{4}</math> | ||
+ | |||
+ | The total area is <math>3\left(1 + \frac{\sqrt{3}}{4}\right) + \frac{\sqrt{3}}{4} = \sqrt{3} + 3</math> | ||
+ | |||
+ | ~mathboy282 | ||
+ | |||
+ | ==Solution 6== | ||
+ | The total area is the sum of the three squares, the three (congruent) obtuse triangles, and the equilateral triangle. The area of the equilateral triangle is <math>\frac{\sqrt{3}}{4}</math> and the area of each square is <math>1</math>. The area of a triangle in general is <math>\frac{1}{2}ab\sin(c)</math> where <math>a</math> and <math>b</math> are two sides and <math>c</math> is the included angle. <math>\angle EAF</math> measures <math>120^{\circ}</math> because <math>\angle EAB</math> and <math>\angle FAC</math> are right, and <math>m\angle CAB=60^{\circ}</math>. So the area of the obtuse triangle is <math>\frac{1}{2}\cdot1\cdot1\cdot\sin\left(120^{\circ}\right)=\frac{\sqrt{3}}{4}</math>. The total area is <math>3\left(\frac{\sqrt{3}}{4}\right)+3\left(1\right)+\frac{\sqrt{3}}{4}=\sqrt{3}+3 \Longrightarrow \boxed{\textbf{(C )}\sqrt{3}+3}</math>. | ||
+ | |||
+ | ~JH. L | ||
+ | |||
+ | |||
+ | ==Solution 7== | ||
+ | Since <math>\angle C=360</math>, <math>\angle GCH=360-90-90-60=120.</math> Applying the Law of Cosines on <math>\angle GCH</math> gives us <math>GH = 1.</math> Since <math>\triangle GCH</math> is isosceles, the perpendicular bisector of <math>\angle C</math> also intersects segment <math>\overline{GH}</math> in its median, which we can call point <math>M.</math> Hence, we can apply the Pythagorean theorem on <math>\triangle CMG</math> or <math>\triangle CMH</math> to get <math>CM = \frac{\sqrt{3}}{4}.</math> We can use this to get the area of the triangle and multiply that by three since the triangles are congruent. The result follows. ~peelybonehead | ||
+ | |||
+ | ==Solution 8== | ||
+ | First, the equilateral triangle has an area of <math>\dfrac{\sqrt{3}}{4}</math>. The three squares have an area of <math>3\times 1=3</math>. | ||
+ | |||
+ | Since <math>\angle C=360</math>, <math>\angle GCH=360-90-90-60=120</math>. | ||
+ | Notice that the three outer isosceles triangles combine to form a new equilateral triangle <math>\triangle IEG</math>. Because it is an equilateral triangle, the two parts of the median separated by the centroid form a ratio of 2:1. Therefore, The altitude of <math>\triangle IEG</math> is <math>\dfrac{3}{2}</math>. That same attitude also creates two 30-60-90 triangles meaning that half of the base of equilateral triangle <math>\triangle IEG</math> is <math>\dfrac{3}{2\sqrt{3}}=\dfrac{3\sqrt{3}}{6}=\dfrac{\sqrt{3}}{2}</math>. Multiplying this by the height yields the area of the triangle to be <math>\dfrac{{3}\sqrt{3}}{4}</math>. Adding all the areas up produces <math>3+\dfrac{\sqrt{3}}{4}+\dfrac{3\sqrt{3}}{4}=3+\dfrac{4\sqrt{3}}{4}=3+\sqrt{3}\implies\boxed{\textbf{(C)}\ 3+\sqrt3}</math>. | ||
+ | |||
==See Also== | ==See Also== | ||
− | {{AMC10 box|year=2014|ab=A|num-b= | + | {{AMC10 box|year=2014|ab=A|num-b=12|num-a=14}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:06, 22 October 2024
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 3
We will use, to find the area of the following triangles. Since , .
The Area of is . Noting, ,
Area of ,
Area of ,
Area of square ABDE = 1,
Therefore the composite area of the entire figure is,
Solution 4
We know that the area is equal to 3*EAF+3*ACGF+ABC. We also know that ACGF and the rest of the squares' area is equal to 1. Therefore the answer is 3*EAF+ABC+3. The only one with "+3" or "3+" is C, our answer. Very unreliable. -Reality Writes
Solution 5
The area of the obtuse triangle is
The total area is
~mathboy282
Solution 6
The total area is the sum of the three squares, the three (congruent) obtuse triangles, and the equilateral triangle. The area of the equilateral triangle is and the area of each square is . The area of a triangle in general is where and are two sides and is the included angle. measures because and are right, and . So the area of the obtuse triangle is . The total area is .
~JH. L
Solution 7
Since , Applying the Law of Cosines on gives us Since is isosceles, the perpendicular bisector of also intersects segment in its median, which we can call point Hence, we can apply the Pythagorean theorem on or to get We can use this to get the area of the triangle and multiply that by three since the triangles are congruent. The result follows. ~peelybonehead
Solution 8
First, the equilateral triangle has an area of . The three squares have an area of .
Since , . Notice that the three outer isosceles triangles combine to form a new equilateral triangle . Because it is an equilateral triangle, the two parts of the median separated by the centroid form a ratio of 2:1. Therefore, The altitude of is . That same attitude also creates two 30-60-90 triangles meaning that half of the base of equilateral triangle is . Multiplying this by the height yields the area of the triangle to be . Adding all the areas up produces .
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.