Difference between revisions of "2014 AMC 8 Problems/Problem 15"
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<math> \textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad\textbf{(E) }150 </math> | <math> \textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad\textbf{(E) }150 </math> | ||
==Solution== | ==Solution== | ||
− | 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 inscribed arc of <math>\angle x=60^{\circ}</math> so <math>m\angle x=30^{\circ}</math>; and the inscribed arc of <math>\angle y=120^{\circ}</math> so <math>m\angle y=60^{\circ}</math>. <math>m\angle x+m\angle y=30+60=</math>\framebox{(C) }< | + | 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 inscribed arc of <math>\angle x=60^{\circ}</math> so <math>m\angle x=30^{\circ}</math>; and the inscribed arc of <math>\angle y=120^{\circ}</math> so <math>m\angle y=60^{\circ}</math>. <math>m\angle x+m\angle y=30+60=</math>\framebox{(C)}<math>90</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 21:50, 2 September 2019
Problem
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
?
Solution
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 of the circle's circumference, each unit central angle measures
. Then, we know that the inscribed arc of
so
; and the inscribed arc of
so
.
\framebox{(C)}
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.