Difference between revisions of "2019 AMC 10B Problems/Problem 8"
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==Problem== | ==Problem== | ||
− | The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region? | + | The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length <math>2</math> and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region? |
− | < | + | <asy> |
+ | pen white = gray(1); | ||
+ | pen gray = gray(0.5); | ||
+ | draw((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle); | ||
+ | fill((0,0)--(2sqrt(3),0)--(2sqrt(3),2sqrt(3))--(0,2sqrt(3))--cycle, gray); | ||
+ | draw((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle); | ||
+ | fill((sqrt(3)-1,0)--(sqrt(3),sqrt(3))--(sqrt(3)+1,0)--cycle, white); | ||
+ | draw((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle); | ||
+ | fill((sqrt(3)-1,2sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3)+1,2sqrt(3))--cycle, white); | ||
+ | draw((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle); | ||
+ | fill((0,sqrt(3)-1)--(sqrt(3),sqrt(3))--(0,sqrt(3)+1)--cycle, white); | ||
+ | draw((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle); | ||
+ | fill((2sqrt(3),sqrt(3)-1)--(sqrt(3),sqrt(3))--(2sqrt(3),sqrt(3)+1)--cycle, white); | ||
+ | </asy> | ||
+ | |||
+ | <math>\textbf{(A) } 4 \qquad \textbf{(B) } 12 - 4\sqrt{3} \qquad \textbf{(C) } 3\sqrt{3}\qquad \textbf{(D) } 4\sqrt{3} \qquad \textbf{(E) } 16 - \sqrt{3}</math> | ||
==Solution== | ==Solution== |
Revision as of 19:35, 17 February 2019
Problem
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?
Solution
We notice that the square can be split into congruent smaller squares with the altitude of the equilateral triangle being the side of the square. Therefore, the area of each shaded part that resides within a square is the total area of the square subtracted from each triangle (Note that it has already been split in half).
When we split an equilateral triangle in half, we get triangles with a relationship. Therefore, we get that the altitude and a side length of a square is .
We can then compute the area of the two triangles using the base-height-area relationship and get .
The area of the small squares is the altitude squared which is . Therefore, the area of the shaded region in each of the four squares is
Since there are four of these squares, we multiply this by to get as our answer. This is choice .
~Awesome2.1 and edited by greersc.
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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.