Difference between revisions of "2024 AMC 8 Problems/Problem 18"
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Three concentric circles centered at <math>O</math> have radii of <math>1</math>, <math>2</math>, and <math>3</math>. Points <math>B</math> and <math>C</math> lie on the largest cirlce. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle <math>BOC</math>, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of <math>\angle{BOC}</math> in degrees? | Three concentric circles centered at <math>O</math> have radii of <math>1</math>, <math>2</math>, and <math>3</math>. Points <math>B</math> and <math>C</math> lie on the largest cirlce. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle <math>BOC</math>, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of <math>\angle{BOC}</math> in degrees? | ||
| − | + | <asy> | |
| + | size(150); | ||
| + | import graph; | ||
| + | draw(circle((0,0),3)); | ||
| + | real radius = 3; | ||
| + | real angleStart = -54; // starting angle of the sector | ||
| + | real angleEnd = 54; // ending angle of the sector | ||
| + | label("$O$",(0,0),W); | ||
| + | pair O = (0, 0); | ||
| + | filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, gray); | ||
| + | filldraw(circle((0,0),2),gray); | ||
| + | filldraw(circle((0,0),1),white); | ||
| + | draw((1.763,2.427)--(0,0)--(1.763,-2.427)); | ||
| + | label("$B$",(1.763,2.427),NE); | ||
| + | label("$C$",(1.763,-2.427),SE); | ||
| + | </asy> | ||
<math>\textbf{(A) } 108\qquad\textbf{(B) } 120\qquad\textbf{(C) } 135\qquad\textbf{(D) } 144\qquad\textbf{(E) } 150</math> | <math>\textbf{(A) } 108\qquad\textbf{(B) } 120\qquad\textbf{(C) } 135\qquad\textbf{(D) } 144\qquad\textbf{(E) } 150</math> | ||
Revision as of 22:11, 26 January 2024
Contents
Problem
Three concentric circles centered at 
have radii of 
, 
, and 
. Points 
and 
lie on the largest cirlce. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle 
, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of 
in degrees?
![[asy] size(150); import graph; draw(circle((0,0),3)); real radius = 3; real angleStart = -54; // starting angle of the sector real angleEnd = 54; // ending angle of the sector label("$O$",(0,0),W); pair O = (0, 0); filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, gray); filldraw(circle((0,0),2),gray); filldraw(circle((0,0),1),white); draw((1.763,2.427)--(0,0)--(1.763,-2.427)); label("$B$",(1.763,2.427),NE); label("$C$",(1.763,-2.427),SE); [/asy]](http://latex.artofproblemsolving.com/4/3/9/439c8e9a56d8540a55cf0ba7052f87041811d6be.png)
Solution
Let 
.
We see that the shaded region is the inner ring plus a sector 
of the outer ring. The area of this in terms of 
is 
. This simplifies to 
.
Also, the unshaded portion is comprised of the smallest circle plus the sector 
of the outer ring. The area of this is 
.
We are told these are equal, therefore 
. Solving for reveals 
.
~MrThinker
Video Solution 1 (super clear!) by Power Solve
Video Solution 2 by Math-X (First fully understand the problem!!!)
Video Solution 3 by SpreadTheMathLove
https://www.youtube.com/watch?v=Svibu3nKB7E
~Math-X
Video Solution 2 by OmegaLearn.org
See Also
| 2024 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
