Difference between revisions of "2019 AMC 10B Problems/Problem 8"
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Since there are <math>4</math> of these squares, we multiply this by <math>4</math> to get <math>4\left(3 - \sqrt{3}\right) = \boxed{\textbf{(B) } 12 - 4\sqrt{3}}</math> as our answer. | Since there are <math>4</math> of these squares, we multiply this by <math>4</math> to get <math>4\left(3 - \sqrt{3}\right) = \boxed{\textbf{(B) } 12 - 4\sqrt{3}}</math> as our answer. | ||
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+ | ==Slightly More Overkill Kite Solution== | ||
+ | We can see that the side length of the square is <math>2\sqrt{3}</math>. Using the Pythagorean theorem, the diagonal of the square <math>\sqrt{12+12}=\sqrt{24}=2\sqrt{6}</math>. Because of this, the height of one of the four shaded kites is <math>\sqrt{6}</math>. Now, we just need to find the length of the length of that kite. Using the Pythagorean theorem, it equals <math>\frac{2\sqrt{3} - 2}{2} \times \sqrt{2} = \sqrt{3} - 1 = \sqrt{6} - \sqrt{2}</math>. Kite area is one half length times width, so the area of one of the four kites is <math>2 \sqrt{6} \times (\sqrt{6}-\sqrt{2}) = 12 - 2\sqrt{12} = 12 - 4\sqrt{3}</math>. | ||
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==See Also== | ==See Also== |
Revision as of 22:21, 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 this smaller square. Therefore, the area of each shaded part that resides within a square is the total area of the square subtracted from each triangle (which has already been split in half).
When we split an equilateral triangle in half, we get two triangles. Therefore, the altitude, which is also the side length of one of the smaller squares, is . We can then compute the area of the two triangles as .
The area of the each small squares is the square of the side length, i.e. . Therefore, the area of the shaded region in each of the four squares is .
Since there are of these squares, we multiply this by to get as our answer.
Slightly More Overkill Kite Solution
We can see that the side length of the square is . Using the Pythagorean theorem, the diagonal of the square . Because of this, the height of one of the four shaded kites is . Now, we just need to find the length of the length of that kite. Using the Pythagorean theorem, it equals . Kite area is one half length times width, so the area of one of the four kites is .
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 |
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