Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2024 AMC 10A Problems/Problem 14"

(→‎Diagram)
(→‎Diagram)
Line 7: Line 7:
  
 
== Diagram ==
 
== Diagram ==
−
[asy]
+
<asy>
 
/* Made by MRENTHUSIASM */
 
/* Made by MRENTHUSIASM */
 
size(250);
 
size(250);
Line 18: Line 18:
 
B = (8*sqrt(3),-8);
 
B = (8*sqrt(3),-8);
 
C = (12*sqrt(3),-8);
 
C = (12*sqrt(3),-8);
−
Label L1 = Label("<math>24</math>", align=(0,0), position=MidPoint, filltype=Fill(0,3,white));
+
Label L1 = Label("$24$", align=(0,0), position=MidPoint, filltype=Fill(0,3,white));
 
fill(A--Arc((12*sqrt(3),4),A,C)--B--cycle,yellow);
 
fill(A--Arc((12*sqrt(3),4),A,C)--B--cycle,yellow);
 
draw(p1^^p2);
 
draw(p1^^p2);
Line 25: Line 25:
 
dot((12*sqrt(3),4),linewidth(4));
 
dot((12*sqrt(3),4),linewidth(4));
 
draw((12*sqrt(3),4)--(12+12*sqrt(3),4));
 
draw((12*sqrt(3),4)--(12+12*sqrt(3),4));
−
label("<math>12</math>",(6+12*sqrt(3),4),1.5S);
+
label("$12$",(6+12*sqrt(3),4),1.5S);
−
[/asy]
+
</asy>

Revision as of 16:58, 8 November 2024

Problem

One side of an equilateral triangle of height $24$ lies on line $\ell$. A circle of radius $12$ is tangent to line $\ ell$ 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 $\ell$ can be written as $a \sqrt{b} - c \pi$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a + b + c$?

$\textbf{(A)}~72\qquad\textbf{(B)}~73\qquad\textbf{(C)}~74\qquad\textbf{(D)}~75\qquad\textbf{(E)}~76$

Diagram

[asy] /* Made by MRENTHUSIASM */ size(250); pair A, B, C; path p1, p2, p3; p1 = scale(16)*polygon(3); p2 = Circle((12*sqrt(3),4),12); A = intersectionpoint(p1,p2); B = (8*sqrt(3),-8); C = (12*sqrt(3),-8); Label L1 = Label("$24$", align=(0,0), position=MidPoint, filltype=Fill(0,3,white)); fill(A--Arc((12*sqrt(3),4),A,C)--B--cycle,yellow); draw(p1^^p2); draw((8*sqrt(3),-8)--(22+8*sqrt(3),-8)); draw((-18,-8)--(-18,16), L=L1, arrow=Arrows(),bar=Bars(15)); dot((12*sqrt(3),4),linewidth(4)); draw((12*sqrt(3),4)--(12+12*sqrt(3),4)); label("$12$",(6+12*sqrt(3),4),1.5S); [/asy]

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2024_AMC_10A_Problems/Problem_14&oldid=231938"