Difference between revisions of "2024 AMC 10A Problems/Problem 14"
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Draw radii to the tangency points, the arc is 60 degrees(The angle that the triangle makes with the line is 120 and because the angles of a quadrilateral sum to 360, we get 120+180+x=360 so 60) | Draw radii to the tangency points, the arc is 60 degrees(The angle that the triangle makes with the line is 120 and because the angles of a quadrilateral sum to 360, we get 120+180+x=360 so 60) | ||
− | Call the bottom vertices B and C (the one closer to the circle be C) and the top vertice A and the tangency point on side of triangle D and on line l E and call center O, triangle ODC is 30-60-90 triangle so CD is <math>4\sqrt{3}</math>. | + | Call the bottom vertices B and C (the one closer to the circle be C) and the top vertice A and the tangency point on the side of triangle D and on line l E and call center O, triangle ODC is 30-60-90 triangle so CD is <math>4\sqrt{3}</math>. |
− | Since we have 2 congruent triangles (<math>\Delta ODC</math> and <math>\Delta OEC</math>), the combined area of both is <math>48\sqrt{3}</math>. and the area of the arc is <math>60*144*\pi/360</math> which is <math>24\pi</math> so the answer is <math> | + | Since we have 2 congruent triangles (<math>\Delta ODC</math> and <math>\Delta OEC</math>), the combined area of both is <math>48\sqrt{3}</math>. and the area of the arc is <math>60*144*\pi/360</math> which is <math>24\pi</math> so the answer is <math>48\sqrt{3}-24\pi</math> so the answer is <math>\textbf{(D)}~75</math> |
− | ASPALAPATI75 | + | ~ASPALAPATI75 |
==See also== | ==See also== | ||
{{AMC10 box|year=2024|ab=A|num-b=13|num-a=15}} | {{AMC10 box|year=2024|ab=A|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 18:13, 8 November 2024
Contents
[hide]Problem
One side of an equilateral triangle of height lies on line . A circle of radius is tangent to line and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line can be written as , where , , and are positive integers and is not divisible by the square of any prime. What is ?
Diagram
Solution 1
Draw radii to the tangency points, the arc is 60 degrees(The angle that the triangle makes with the line is 120 and because the angles of a quadrilateral sum to 360, we get 120+180+x=360 so 60)
Call the bottom vertices B and C (the one closer to the circle be C) and the top vertice A and the tangency point on the side of triangle D and on line l E and call center O, triangle ODC is 30-60-90 triangle so CD is . Since we have 2 congruent triangles ( and ), the combined area of both is . and the area of the arc is which is so the answer is so the answer is
~ASPALAPATI75
See also
2024 AMC 10A (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 |
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