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2007 AMC 10B Problems/Problem 4: Difference between revisions

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==Problem 4==
==Problem ==


The point <math>O</math> is the center of the circle circumscribed about <math>\triangle ABC,</math> with <math>\angle BOC=120^\circ</math> and <math>\angle AOB=140^\circ,</math> as shown. What is the degree measure of <math>\angle ABC?</math>
The point <math>O</math> is the center of the circle circumscribed about <math>\triangle ABC,</math> with <math>\angle BOC=120^\circ</math> and <math>\angle AOB=140^\circ,</math> as shown. What is the degree measure of <math>\angle ABC?</math>
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\angle BOC + \angle AOB + \angle AOC &= 360\\
\angle BOC + \angle AOB + \angle AOC &= 360\\
120 + 140 + \angle AOC &= 360\\
120 + 140 + \angle AOC &= 360\\
\angle AOC &= 120.
\angle AOC &= 100.
\end{align*}</cmath>
\end{align*}</cmath>


Therefore, the measure of <math>\text{arc}AC</math> is also <math>120^\circ.</math> Since the measure of an [[inscribed angle]] is equal to half the measure of the arc it intercepts, <math>\angle ABC = \boxed{\textbf{(E)} 60}</math>
Therefore, the measure of <math>\text{arc}AC</math> is also <math>100^\circ.</math> Since the measure of an [[inscribed angle]] is equal to half the measure of the arc it intercepts, <math>\angle ABC = \boxed{\textbf{(D)} 50}</math>
 
== See Also ==
 
{{AMC10 box|year=2007|ab=B|num-b=3|num-a=5}}
{{MAA Notice}}

Latest revision as of 11:19, 4 July 2013

Problem

The point $O$ is the center of the circle circumscribed about $\triangle ABC,$ with $\angle BOC=120^\circ$ and $\angle AOB=140^\circ,$ as shown. What is the degree measure of $\angle ABC?$

$\textbf{(A) } 35 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 50 \qquad\textbf{(E) } 60$

Solution

Because all the central angles of a circle add up to $360^\circ,$

\begin{align*} \angle BOC + \angle AOB + \angle AOC &= 360\\ 120 + 140 + \angle AOC &= 360\\ \angle AOC &= 100. \end{align*}

Therefore, the measure of $\text{arc}AC$ is also $100^\circ.$ Since the measure of an inscribed angle is equal to half the measure of the arc it intercepts, $\angle ABC = \boxed{\textbf{(D)} 50}$

See Also

2007 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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 AMC 10 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America.

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