Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2014 AMC 10A Problems/Problem 13"

(→‎Solution 4)
 
(6 intermediate revisions by 4 users not shown)
Line 36: Line 36:
 
label("$E$",E,W);
 
label("$E$",E,W);
 
label("$F$",F,E);
 
label("$F$",F,E);
−
label("$G$",G,E);
+
label("$G$",G,SE);
 
label("$H$",H,SE);
 
label("$H$",H,SE);
 
label("$I$",I,SW);
 
label("$I$",I,SW);
Line 129: Line 129:
 
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.
 
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
 
-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==

Latest revision as of 22:06, 22 October 2024

Problem

Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?

[asy] import graph; size(6cm); 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; 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); 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,SE); label("$H$",H,SE); label("$I$",I,SW); [/asy]

$\textbf{(A)}\ \dfrac{12+3\sqrt3}4\qquad\textbf{(B)}\ \dfrac92\qquad\textbf{(C)}\ 3+\sqrt3\qquad\textbf{(D)}\ \dfrac{6+3\sqrt3}2\qquad\textbf{(E)}\ 6$

Solution 1

The area of the equilateral triangle is $\dfrac{\sqrt{3}}{4}$. The area of the three squares is $3\times 1=3$.

Since $\angle C=360$, $\angle GCH=360-90-90-60=120$.

Dropping an altitude from $C$ to $GH$ allows to create a $30-60-90$ triangle since $\triangle GCH$ is isosceles. This means that the height of $\triangle GCH$ is $\dfrac{1}{2}$ and half the length of $GH$ is $\dfrac{\sqrt{3}}{2}$. Therefore, the area of each isosceles triangle is $\dfrac{1}{2}\times\dfrac{\sqrt{3}}{2}=\dfrac{\sqrt{3}}{4}$. Multiplying by $3$ yields $\dfrac{3\sqrt{3}}{4}$ for all three isosceles triangles.

Therefore, the total area is $3+\dfrac{\sqrt{3}}{4}+\dfrac{3\sqrt{3}}{4}=3+\dfrac{4\sqrt{3}}{4}=3+\sqrt{3}\implies\boxed{\textbf{(C)}\ 3+\sqrt3}$.

Solution 2

As seen in the previous solution, segment $GH$ is $\sqrt{3}$. Think of the picture as one large equilateral triangle, $\triangle{JKL}$ with the sides of $2\sqrt{3}+1$, by extending $EF$, $GH$, and $DI$ to points $J$, $K$, and $L$, respectively. This makes the area of $\triangle{JKL}$ $\dfrac{\sqrt{3}}{4}(2\sqrt{3}+1)^2=\dfrac{12+13\sqrt{3}}{4}$.

[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 $\triangle{DIJ}$, $\triangle{EFK}$, and $\triangle{GHL}$ have sides of $\sqrt{3}$, so their total area is $3(\dfrac{\sqrt{3}}{4}(\sqrt{3})^2)=\dfrac{9\sqrt{3}}{4}$.

Now, you subtract their total area from the area of $\triangle{JKL}$:

$\dfrac{12+13\sqrt{3}}{4}-\dfrac{9\sqrt{3}}{4}=\dfrac{12+13\sqrt{3}-9\sqrt{3}}{4}=\dfrac{12+4\sqrt{3}}{4}=3+\sqrt{3}\implies\boxed{\textbf{(C)}\ 3+\sqrt3}$

Solution 3

We will use, $\frac{1}{2}ab\sin x$ to find the area of the following triangles. Since $\angle A=360$, $\angle EAF=360-90-90-60=120$.

The Area of $\triangle AEF$ is $\frac{1}{2} \cdot 1 \cdot 1 \cdot \sin(120)$. Noting, $\sin (2x) = 2\sin (x)\cos (x)$,

Area of $\triangle AEF = \frac{1}{2} \cdot 1 \cdot 1 \cdot 2 \cdot \sin(60) \cdot \cos(60) = \dfrac{\sqrt{3}}{4}$,

Area of $\triangle ABC = \frac{1}{2} \cdot 1 \cdot 1 \cdot \sin(60) = \dfrac{\sqrt{3}}{4}$,

Area of square ABDE = 1,

Therefore the composite area of the entire figure is, \[3 \cdot [\triangle AEF] + [\triangle ABC] + 3 \cdot [ABDE] = 3 \dfrac{\sqrt{3}}{4} + \dfrac{\sqrt{3}}{4} + 3 \cdot 1 = 4 \dfrac{\sqrt{3}}{4} + 3 = \sqrt{3} + 3 \implies\boxed{\textbf{(C)}\ 3+\sqrt3}\]


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

$\angle{AEF} = 180- \angle{BAC} = 120$

The area of the obtuse triangle is $\frac{1}{2}\sin{120} = \frac{\sqrt{3}}{4}$

The total area is $3\left(1 + \frac{\sqrt{3}}{4}\right) + \frac{\sqrt{3}}{4} = \sqrt{3} + 3$

~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 $\frac{\sqrt{3}}{4}$ and the area of each square is $1$. The area of a triangle in general is $\frac{1}{2}ab\sin(c)$ where $a$ and $b$ are two sides and $c$ is the included angle. $\angle EAF$ measures $120^{\circ}$ because $\angle EAB$ and $\angle FAC$ are right, and $m\angle CAB=60^{\circ}$. So the area of the obtuse triangle is $\frac{1}{2}\cdot1\cdot1\cdot\sin\left(120^{\circ}\right)=\frac{\sqrt{3}}{4}$. The total area is $3\left(\frac{\sqrt{3}}{4}\right)+3\left(1\right)+\frac{\sqrt{3}}{4}=\sqrt{3}+3 \Longrightarrow \boxed{\textbf{(C )}\sqrt{3}+3}$.

~JH. L


Solution 7

Since $\angle C=360$, $\angle GCH=360-90-90-60=120.$ Applying the Law of Cosines on $\angle GCH$ gives us $GH = 1.$ Since $\triangle GCH$ is isosceles, the perpendicular bisector of $\angle C$ also intersects segment $\overline{GH}$ in its median, which we can call point $M.$ Hence, we can apply the Pythagorean theorem on $\triangle CMG$ or $\triangle CMH$ to get $CM = \frac{\sqrt{3}}{4}.$ 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 $\dfrac{\sqrt{3}}{4}$. The three squares have an area of $3\times 1=3$.

Since $\angle C=360$, $\angle GCH=360-90-90-60=120$. Notice that the three outer isosceles triangles combine to form a new equilateral triangle $\triangle IEG$. 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 $\triangle IEG$ is $\dfrac{3}{2}$. That same attitude also creates two 30-60-90 triangles meaning that half of the base of equilateral triangle $\triangle IEG$ is $\dfrac{3}{2\sqrt{3}}=\dfrac{3\sqrt{3}}{6}=\dfrac{\sqrt{3}}{2}$. Multiplying this by the height yields the area of the triangle to be $\dfrac{{3}\sqrt{3}}{4}$. Adding all the areas up produces $3+\dfrac{\sqrt{3}}{4}+\dfrac{3\sqrt{3}}{4}=3+\dfrac{4\sqrt{3}}{4}=3+\sqrt{3}\implies\boxed{\textbf{(C)}\ 3+\sqrt3}$.

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2014_AMC_10A_Problems/Problem_13&oldid=230380"