Difference between revisions of "2007 AMC 12B Problems/Problem 3"
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==Problem== | ==Problem== | ||
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The point <math>O</math> is the center of the circle circumscribed about triangle <math>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 triangle <math>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>? | ||
− | + | [[Image:2007_12B_AMC-3.png]] | |
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+ | <math>\mathrm {(A)} 35 \qquad \mathrm {(B)} 40 \qquad \mathrm {(C)} 45 \qquad \mathrm {(D)} 50 \qquad \mathrm {(E)} 60</math> | ||
==Solution== | ==Solution== | ||
− | + | Since triangles <math>ABO</math> and <math>BOC</math> are isosceles, <math>\angle ABO=20^o</math> and <math>\angle OBC=30^o</math>. Therefore, <math>\angle ABC=50^o</math>, or <math>\mathrm{(D)}</math>. | |
− | <math>\angle | ||
− | <math>\angle ABC=50 \ | + | ==Alternative Solution== |
+ | <math>\angle AOC = 100^{\circ} \implies \angle ABC =\frac{\angle AOC}{2} =50^{ \circ},</math> or <math>\mathrm{(D)}.</math> | ||
==See Also== | ==See Also== | ||
+ | {{AMC12 box|year=2007|ab=B|num-b=2|num-a=4}} | ||
− | {{ | + | [[Category:Introductory Algebra Problems]] |
+ | {{MAA Notice}} |
Latest revision as of 09:39, 27 February 2022
Problem
The point is the center of the circle circumscribed about triangle , with and , as shown. What is the degree measure of ?
Solution
Since triangles and are isosceles, and . Therefore, , or .
Alternative Solution
or
See Also
2007 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 2 |
Followed by Problem 4 |
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 12 Problems and Solutions |
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