Difference between revisions of "2014 AMC 8 Problems/Problem 15"

Latest revision as of 16:40, 30 January 2025

Problem

The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$ ?

$[asy] size(230); defaultpen(linewidth(0.65)); pair O=origin; pair[] circum = new pair[12]; string[] let = {"$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$",O,dir(60)); [/asy]$

$\textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad\textbf{(E) }150$

Solution 1

The measure of an inscribed angle is half the measure of its corresponding central angle. Since each unit arc is $\frac{1}{12}$ of the circle's circumference, each unit central angle measures $\frac{360}{12}^{\circ}=30^{\circ}$ . From this, $\angle EOG = 60^{\circ}$ , so $x = 30^{\circ}$ . Also, $\angle AOI = 120^{\circ}$ , so $y = 60^{\circ}$ . The number of degrees in the sum of both angles is $30 + 60 = \boxed{(C)\ 90}.$

Solution 2

Since $\triangle AOE$ is isosceles and $\angle AOE = \frac{4}{12} \cdot 360^{\circ} = 120^{\circ}$ , $x = 30^{\circ}$ . Since $\triangle GOI$ is isosceles and $\angle GOI = \frac{2}{12} \cdot 360^{\circ} = 60^{\circ}$ , $x = 60^{\circ}$ . The number of degrees in the sum of both angles is $30+60 = \boxed{(C)\ 90}$ .

Video Solution 1

https://youtu.be/3QHH9xV-QDw

~Education, the Study of Everything

Video Solution 2

https://www.youtube.com/watch?v=qseG63LK4AU ~David

Video Solution 3

https://youtu.be/aZhjhb3mMfg ~savannahsolver

Video Solution 4

https://youtu.be/abSgjn4Qs34?t=3242

@@ Line 1: / Line 1: @@
-==Problem==
+== Problem ==
 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>?
@@ Line 23: / Line 24: @@
 <math> \textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad\textbf{(E) }150 </math>
-==Video Solution (CREATIVE THINKING)==
+== Solution 1 ==
+The measure of an inscribed angle is half the measure of its corresponding central angle. Since each unit arc is <math>\frac{1}{12}</math> of the circle's circumference, each unit central angle measures <math>\frac{360}{12}^{\circ}=30^{\circ}</math>. From this, <math>\angle EOG = 60^{\circ}</math>, so <math>x = 30^{\circ}</math>. Also, <math>\angle AOI = 120^{\circ}</math>, so <math>y = 60^{\circ}</math>. The number of degrees in the sum of both angles is <math>30 + 60 = \boxed{(C)\ 90}.</math>
+== Solution 2 ==
+Since <math>\triangle AOE</math> is isosceles and <math>\angle AOE = \frac{4}{12} \cdot 360^{\circ} = 120^{\circ}</math>, <math>x = 30^{\circ}</math>. Since <math>\triangle GOI</math> is isosceles and <math>\angle GOI = \frac{2}{12} \cdot 360^{\circ} = 60^{\circ}</math>, <math>x = 60^{\circ}</math>. The number of degrees in the sum of both angles is <math>30+60 = \boxed{(C)\ 90}</math>.
+== Video Solution 1 ==
 https://youtu.be/3QHH9xV-QDw
 ~Education, the Study of Everything
+== Video Solution 2 ==
+https://www.youtube.com/watch?v=qseG63LK4AU
+~David
+== Video Solution 3 ==
-==Video Solution==
+https://youtu.be/aZhjhb3mMfg
-https://www.youtube.com/watch?v=qseG63LK4AU   ~David
+~savannahsolver
-https://youtu.be/aZhjhb3mMfg ~savannahsolver
+== Video Solution 4 ==
-==Video Solution ==
 https://youtu.be/abSgjn4Qs34?t=3242
-==Solution==
+== See Also ==
-For this problem, it is useful to know that the measure of an inscribed angle is half the measure of its corresponding central angle. Since each unit arc is <math>\frac{1}{12}</math> of the circle's circumference, each unit central angle measures <math>\left( \frac{360}{12} \right) ^{\circ}=30^{\circ}</math>. Then, we know that the central angle of x = <math>60</math>, so inscribed angle = <math>30</math>. Also, central angle of y = <math>120</math>, so inscribed angle = <math>60</math>. Summing both inscribed angles gives <math>30 + 60 = \boxed{(C)\ 90}.</math>
-==See Also==
 {{AMC8 box|year=2014|num-b=14|num-a=16}}
 {{MAA Notice}}

2014 AMC 8 (Problems • Answer Key • Resources)
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