Difference between revisions of "2014 AMC 8 Problems/Problem 15"
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| − | + | The circumference of the circle with center <math>O</math> is divided into <math>12</math> equal arcs, marked the letters <math>A</math> through <math>L</math> as seen below. What is the number of degrees in the sum of the angles <math>x</math> and <math>y</math>? | |
| + | |||
| + | <asy> | ||
| + | size(230); | ||
| + | defaultpen(linewidth(0.65)); | ||
| + | pair O=origin; | ||
| + | pair[] circum = new pair[12]; | ||
| + | string[] let = {"$A$","$B$","$C$","$D$","$E$","$F$","$G$","$H$","$I$","$J$","$K$","$L$"}; | ||
| + | draw(unitcircle); | ||
| + | for(int i=0;i<=11;i=i+1) | ||
| + | { | ||
| + | circum[i]=dir(120-30*i); | ||
| + | dot(circum[i],linewidth(2.5)); | ||
| + | label(let[i],circum[i],2*dir(circum[i])); | ||
| + | } | ||
| + | draw(O--circum[4]--circum[0]--circum[6]--circum[8]--cycle); | ||
| + | label("$x$",circum[0],2.75*(dir(circum[0]--circum[4])+dir(circum[0]--circum[6]))); | ||
| + | label("$y$",circum[6],1.75*(dir(circum[6]--circum[0])+dir(circum[6]--circum[8]))); | ||
| + | label("$O$",O,dir(60)); | ||
| + | </asy> | ||
| + | |||
| + | <math> \textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad\textbf{(E) }150 </math> | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2014|num-b=14|num-a=16}} | {{AMC8 box|year=2014|num-b=14|num-a=16}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 19:45, 26 November 2014
The circumference of the circle with center 
is divided into 
equal arcs, marked the letters 
through 
as seen below. What is the number of degrees in the sum of the angles 
and 
?
![[asy] size(230); defaultpen(linewidth(0.65)); pair O=origin; pair[] circum = new pair[12]; string[] let = {"$A$","$B$","$C$","$D$","$E$","$F$","$G$","$H$","$I$","$J$","$K$","$L$"}; draw(unitcircle); for(int i=0;i<=11;i=i+1) { circum[i]=dir(120-30*i); dot(circum[i],linewidth(2.5)); label(let[i],circum[i],2*dir(circum[i])); } draw(O--circum[4]--circum[0]--circum[6]--circum[8]--cycle); label("$x$",circum[0],2.75*(dir(circum[0]--circum[4])+dir(circum[0]--circum[6]))); label("$y$",circum[6],1.75*(dir(circum[6]--circum[0])+dir(circum[6]--circum[8]))); label("$O$",O,dir(60)); [/asy]](http://latex.artofproblemsolving.com/3/7/1/371f42726ae702ba57a0002d787f0e3c62767ec2.png)
See Also
| 2014 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 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. 
