Difference between revisions of "2016 AMC 10B Problems/Problem 23"
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From this, we know that <math>CF=2</math>. We also know that the combined heights of the trapezoids is <math>\frac{\sqrt 3}3</math>, since <math>\overline{ZW}</math> and <math>\overline{YX}</math> are equally spaced, and the height of each of the trapezoids is <math>\frac{\sqrt 3}6</math>. From this, we know <math>\overline{ZW}</math> and <math>\overline{YX}</math> are each <math>\frac 13</math> of the way from <math>\overline{CF}</math> to <math>\overline{DE}</math> and <math>\overline{AB}</math>, respectively. We know that these are both equal to <math>\frac 53</math>. | From this, we know that <math>CF=2</math>. We also know that the combined heights of the trapezoids is <math>\frac{\sqrt 3}3</math>, since <math>\overline{ZW}</math> and <math>\overline{YX}</math> are equally spaced, and the height of each of the trapezoids is <math>\frac{\sqrt 3}6</math>. From this, we know <math>\overline{ZW}</math> and <math>\overline{YX}</math> are each <math>\frac 13</math> of the way from <math>\overline{CF}</math> to <math>\overline{DE}</math> and <math>\overline{AB}</math>, respectively. We know that these are both equal to <math>\frac 53</math>. | ||
− | We find the area of each of the trapezoids, which both happen to be <math>\frac{11}6 \cdot \frac{\sqrt 3}6=\frac{11\sqrt 3}{36}</math>, and the combined area is <math>\frac{11\sqrt 3}{18 | + | We find the area of each of the trapezoids, which both happen to be <math>\frac{11}6 \cdot \frac{\sqrt 3}6=\frac{11\sqrt 3}{36}</math>, and the combined area is <math>\frac{11\sqrt 3}{18}</math>. |
We find that <math>\dfrac{\frac{11\sqrt 3}{18}}{\frac{3\sqrt 3}2}</math> is equal to <math>\frac{22}{54}=\boxed{\textbf{(C)}\ \frac{11}{27}}</math>. | We find that <math>\dfrac{\frac{11\sqrt 3}{18}}{\frac{3\sqrt 3}2}</math> is equal to <math>\frac{22}{54}=\boxed{\textbf{(C)}\ \frac{11}{27}}</math>. | ||
− | + | ==Solution 2== | |
− | |||
− | |||
− | ==Solution 2 | ||
<asy> | <asy> | ||
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Thus, the answer is <math>\frac{22}{54}=\boxed{\textbf{(C)}\ \frac{11}{27}}</math>. | Thus, the answer is <math>\frac{22}{54}=\boxed{\textbf{(C)}\ \frac{11}{27}}</math>. | ||
+ | |||
+ | ==Solution 3 (Similar Triangles)== | ||
+ | <asy> | ||
+ | pair A,B,C,D,E,F,W,X,Y,Z; | ||
+ | A=(0,0); | ||
+ | B=(1,0); | ||
+ | C=(3/2,sqrt(3)/2); | ||
+ | D=(1,sqrt(3)); | ||
+ | E=(0,sqrt(3)); | ||
+ | F=(-1/2,sqrt(3)/2); | ||
+ | W=(4/3,2sqrt(3)/3); | ||
+ | X=(4/3,sqrt(3)/3); | ||
+ | Y=(-1/3,sqrt(3)/3); | ||
+ | Z=(-1/3,2sqrt(3)/3); | ||
+ | pair G = (0.5, sqrt(3)*3/2); | ||
+ | draw(A--B--C--D--E--F--cycle); | ||
+ | draw(W--Z); | ||
+ | draw(X--Y); | ||
+ | draw(E--G--D); | ||
+ | draw(F--C); | ||
+ | |||
+ | label("$A$",A,SW); | ||
+ | label("$B$",B,SE); | ||
+ | label("$C$",C,ESE); | ||
+ | label("$D$",D,NE); | ||
+ | label("$E$",E,NW); | ||
+ | label("$F$",F,WSW); | ||
+ | label("$W$",W,ENE); | ||
+ | label("$X$",X,ESE); | ||
+ | label("$Y$",Y,WSW); | ||
+ | label("$Z$",Z,WNW); | ||
+ | label("$G$",G,N); | ||
+ | </asy> | ||
+ | Extend <math>\overline{EF}</math> and <math>\overline{CD}</math> to meet at point <math>G</math>, as shown in the diagram. Then <math>\triangle GZW \sim \triangle GFC</math>. Then <math>[GZW] = \left(\dfrac53\right)^2[GED]</math> and <math>[GFC] = 2^2[GED]</math>. Subtracting <math>[GED]</math>, we find that <math>[EDWZ] = \dfrac{16}{9}[GED]</math> and <math>[EDCF] = 3[GED]</math>. Subtracting again, we find that <cmath>[ZWCF] = [EDCF] - [EDWZ] = \dfrac{11}{9}[GED].</cmath>Finally, <cmath>\dfrac{[WCXYFZ]}{[ABCDEF]} = \dfrac{[ZWCF]}{[EDCF]} = \dfrac{\dfrac{11}{9}[GED]}{3[GED]} = \textbf{(C) } \dfrac{11}{27}.</cmath> | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2016|ab=B|num-b=22|num-a=24}} | {{AMC10 box|year=2016|ab=B|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 02:38, 9 August 2020
Problem
In regular hexagon , points , , , and are chosen on sides , , , and respectively, so lines , , , and are parallel and equally spaced. What is the ratio of the area of hexagon to the area of hexagon ?
Solution 1
We draw a diagram to make our work easier:
Assume that is of length . Therefore, the area of is . To find the area of , we draw , and find the area of the trapezoids and .
From this, we know that . We also know that the combined heights of the trapezoids is , since and are equally spaced, and the height of each of the trapezoids is . From this, we know and are each of the way from to and , respectively. We know that these are both equal to .
We find the area of each of the trapezoids, which both happen to be , and the combined area is .
We find that is equal to .
Solution 2
First, like in the first solution, split the large hexagon into 6 equilateral triangles. Each equilateral triangle can be split into three rows of smaller equilateral triangles. The first row will have one triangle, the second three, the third five. Once you have drawn these lines, it's just a matter of counting triangles. There are small triangles in hexagon , and small triangles in the whole hexagon.
Thus, the answer is .
Solution 3 (Similar Triangles)
Extend and to meet at point , as shown in the diagram. Then . Then and . Subtracting , we find that and . Subtracting again, we find that Finally,
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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