2020 AMC 10B Problems/Problem 14: Difference between revisions
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==Solution== | ==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)); | |||
dot((2,0)); | |||
dot((0.5,a)); | |||
label("2",(z,c),NE); | |||
label("A",(2,0),S); | |||
label("B",(0.5,a),SW); | |||
label("C",(1.5,a),NE); | |||
label("1",(1.5,0),S); | |||
</asy> | |||
==Video Solution== | ==Video Solution== | ||
Revision as of 17:40, 7 February 2020
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)); dot((2,0)); dot((0.5,a)); label("2",(z,c),NE); label("A",(2,0),S); label("B",(0.5,a),SW); label("C",(1.5,a),NE); label("1",(1.5,0),S); [/asy]](http://latex.artofproblemsolving.com/9/a/7/9a7863b9af65a052a3848b76ce4fa1d7cddc8718.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 | ||
These problems are copyrighted © by the Mathematical Association of America. 
