Difference between revisions of "1994 AHSME Problems/Problem 29"
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Let the center of this circle be <math>O</math>, <math>\angle BOC = \theta</math>, the radius of <math>\odot O</math> be <math>r</math>. | Let the center of this circle be <math>O</math>, <math>\angle BOC = \theta</math>, the radius of <math>\odot O</math> be <math>r</math>. | ||
− | + | By the definition of radian, <math>\theta = </math> | |
+ | <math>\overarc{BC}</math> | ||
+ | <math>/r=1</math> | ||
+ | |||
<math>\angle BAC=\frac{\angle BOC}{2} = \frac12</math> | <math>\angle BAC=\frac{\angle BOC}{2} = \frac12</math> | ||
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<math>\sin \angle BAD = \frac{BD}{AB}</math>, <math>\sin \frac{\angle BAC}{2} = \frac{\frac{BC}{2}}{AB}</math> | <math>\sin \angle BAD = \frac{BD}{AB}</math>, <math>\sin \frac{\angle BAC}{2} = \frac{\frac{BC}{2}}{AB}</math> | ||
− | <math>\frac{BC}{AB}=2 \sin \frac{\frac12}{2} = \sin \frac14</math> | + | <math>\frac{BC}{AB}=2 \sin \frac{\frac12}{2} = 2\sin \frac14</math> |
− | <math>\frac{AB}{BC}= \boxed{\textbf{(A) } \frac{1}{2} \csc \frac{1}{4} }</math> | + | <math>\frac{AB}{BC}= \frac{1}{2\sin \frac14} = \boxed{\textbf{(A) } \frac{1}{2} \csc \frac{1}{4} }</math> |
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] |
Latest revision as of 08:19, 28 September 2023
Contents
Problem
Points and on a circle of radius are situated so that , and the length of minor arc is . If angles are measured in radians, then
Solution 1
First note that arc length equals , where is the central angle in radians. Call the center of the circle . Then radian because the minor arc has length . Since is isosceles, . We use the Law of Cosines to find that Using half-angle formulas, we have that this ratio simplifies to
Solution 2
Let the center of this circle be , , the radius of be .
By the definition of radian,
,
See Also
1994 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 28 |
Followed by Problem 30 | |
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 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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