Difference between revisions of "2010 AMC 10B Problems/Problem 6"
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label("$O$", O, S); | label("$O$", O, S); | ||
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| + | ==Video Solution== | ||
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| + | ~Education, the Study of Everything | ||
==Video Solution== | ==Video Solution== | ||
Latest revision as of 16:00, 1 August 2022
Problem
A circle is centered at 
, 
is a diameter and 
is a point on the circle with 
. What is the degree measure of 
?
Solution 1
Assuming we do not already know an inscribed angle is always half of its central angle, we will try a different approach. Since is the center, 
and 
are radii and they are congruent. Thus, 
is an isosceles triangle. Also, note that 
and 
are supplementary, then 
. Since is isosceles, then 
. They also sum to 
, so each angle is 
.
Solution 2 (Alcumus)
Note that 
. Because triangle 
is isosceles, 
.
![[asy] import graph; unitsize(2 cm); pair O, A, B, C; O = (0,0); A = (-1,0); B = (1,0); C = dir(50); draw(Circle(O,1)); draw(B--A--C--O); label("$A$", A, W); label("$B$", B, E); label("$C$", C, NE); label("$O$", O, S); [/asy]](http://latex.artofproblemsolving.com/a/a/b/aab4fe9f0a401fcda884f1328c442a93c5cf59bb.png)
Video Solution
~Education, the Study of Everything
Video Solution
~IceMatrix
See Also
| 2010 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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, as part of the American Mathematics Competitions. 
