Difference between revisions of "2020 AMC 10B Problems/Problem 14"
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draw(arc((2,y),1,180,360)); | draw(arc((2,y),1,180,360)); | ||
draw(arc((2,0),1,0,180)); | draw(arc((2,0),1,0,180)); | ||
| − | + | pair G,H,I,J,K; | |
| − | + | G = (2,0); | |
| + | H = (2.5,a); | ||
| + | I = (1.5,a); | ||
| + | J = (1,0); | ||
| + | K = (3,0); | ||
| + | dot(G); | ||
| + | dot(H); | ||
| + | dot(I); | ||
| + | dot(J); | ||
| + | dot(K); | ||
label("2",(z,c),NE); | label("2",(z,c),NE); | ||
| − | |||
| − | |||
| − | |||
label("1",(1.5,0),S); | label("1",(1.5,0),S); | ||
| + | label("1",(2.5,0),S); | ||
| + | label("1",(1.25,0.5a),SE); | ||
| + | label("1",(2.75,0.5a),SW); | ||
| + | label("1",(2.75,0.5a),SW); | ||
| + | label("$60^\circ$",anglemark(H,G,I),2*N); | ||
| + | draw(anglemark(H,G,I,8),blue); | ||
| + | draw(G--J--I--G); | ||
| + | draw(G--H--K--G); | ||
</asy> | </asy> | ||
Revision as of 17:56, 7 February 2020
Contents
Problem
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region — inside the hexagon but outside all of the semicircles?
![[asy] real x=sqrt(3); real y=2sqrt(3); real z=3.5; real a=x/2; real b=0.5; real c=3a; pair A, B, C, D, E, F; A = (1,0); B = (3,0); C = (4,x); D = (3,y); E = (1,y); F = (0,x); fill(A--B--C--D--E--F--A--cycle,grey); fill(arc((2,0),1,0,180)--cycle,white); fill(arc((2,y),1,180,360)--cycle,white); fill(arc((z,a),1,60,240)--cycle,white); fill(arc((b,a),1,300,480)--cycle,white); fill(arc((b,c),1,240,420)--cycle,white); fill(arc((z,c),1,120,300)--cycle,white); draw(A--B--C--D--E--F--A); draw(arc((z,c),1,120,300)); draw(arc((b,c),1,240,420)); draw(arc((b,a),1,300,480)); draw(arc((z,a),1,60,240)); draw(arc((2,y),1,180,360)); draw(arc((2,0),1,0,180)); label("2",(z,c),NE); [/asy]](http://latex.artofproblemsolving.com/5/1/f/51f2f63151f092f8afd58fa9b0f466ee4a96fbf0.png)
Solution
![[asy] real x=sqrt(3); real y=2sqrt(3); real z=3.5; real a=x/2; real b=0.5; real c=3a; pair A, B, C, D, E, F; A = (1,0); B = (3,0); C = (4,x); D = (3,y); E = (1,y); F = (0,x); fill(A--B--C--D--E--F--A--cycle,grey); fill(arc((2,0),1,0,180)--cycle,white); fill(arc((2,y),1,180,360)--cycle,white); fill(arc((z,a),1,60,240)--cycle,white); fill(arc((b,a),1,300,480)--cycle,white); fill(arc((b,c),1,240,420)--cycle,white); fill(arc((z,c),1,120,300)--cycle,white); draw(A--B--C--D--E--F--A); draw(arc((z,c),1,120,300)); draw(arc((b,c),1,240,420)); draw(arc((b,a),1,300,480)); draw(arc((z,a),1,60,240)); draw(arc((2,y),1,180,360)); draw(arc((2,0),1,0,180)); pair G,H,I,J,K; G = (2,0); H = (2.5,a); I = (1.5,a); J = (1,0); K = (3,0); dot(G); dot(H); dot(I); dot(J); dot(K); label("2",(z,c),NE); label("1",(1.5,0),S); label("1",(2.5,0),S); label("1",(1.25,0.5a),SE); label("1",(2.75,0.5a),SW); label("1",(2.75,0.5a),SW); label("$60^\circ$",anglemark(H,G,I),2*N); draw(anglemark(H,G,I,8),blue); draw(G--J--I--G); draw(G--H--K--G); [/asy]](http://latex.artofproblemsolving.com/a/d/e/adedc69fc4419e206c6e950980ba57e29445016f.png)
Video Solution
~IceMatrix
See Also
| 2020 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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. 
