Difference between revisions of "2005 AMC 10A Problems/Problem 12"
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The figure shown is called a ''trefoil'' and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length <math>2</math>? | The figure shown is called a ''trefoil'' and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length <math>2</math>? | ||
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unitsize(1.5cm); | unitsize(1.5cm); | ||
defaultpen(linewidth(.8pt)+fontsize(12pt)); | defaultpen(linewidth(.8pt)+fontsize(12pt)); | ||
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draw((0.1,-0.12)--(1,-0.12),EndArrow(HookHead,4),EndBar); | draw((0.1,-0.12)--(1,-0.12),EndArrow(HookHead,4),EndBar); | ||
draw((-0.1,-0.12)--(-1,-0.12),EndArrow(HookHead,4),EndBar); | draw((-0.1,-0.12)--(-1,-0.12),EndArrow(HookHead,4),EndBar); | ||
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<math> \mathrm{(A) \ } \frac{1}{3}\pi+\frac{\sqrt{3}}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\pi\qquad \mathrm{(C) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{4}\qquad \mathrm{(D) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{3}\qquad \mathrm{(E) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{2} </math> | <math> \mathrm{(A) \ } \frac{1}{3}\pi+\frac{\sqrt{3}}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\pi\qquad \mathrm{(C) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{4}\qquad \mathrm{(D) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{3}\qquad \mathrm{(E) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{2} </math> |
Revision as of 10:39, 12 July 2021
Problem
The figure shown is called a trefoil and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length ?
Solution
The area of the trefoil is equal to the area of a small equilateral triangle plus the area of four sectors with a radius of minus the area of a small equilateral triangle.
This is equivalent to the area of four sectors with a radius of .
So the answer is:
See also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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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