1985 AHSME Problems/Problem 22: Difference between revisions
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==Problem== | ==Problem== | ||
In a circle with center <math> O </math>, <math> AD </math> is a [[diameter]], <math> ABC </math> is a [[chord]], <math> BO=5 </math> | In a circle with center <math> O </math>, <math> AD </math> is a [[diameter]], <math> ABC </math> is a [[chord]], <math> BO=5 </math> and <math> \angle ABO= \ \stackrel{\frown}{CD} \ =60^\circ </math>. Then the length of <math> BC </math> is | ||
<asy> | <asy> | ||
Revision as of 00:08, 3 April 2018
Problem
In a circle with center 
, 
is a diameter, 
is a chord, 
and 
. Then the length of 
is
![[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C); draw(C--A--D^^B--O^^Circle(O,1)); pair point=O; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$O$", O, dir(305)); label("$5$", B--O, dir(O--B)*dir(90)); label("$60^\circ$", dir(185), dir(185)); label("$60^\circ$", B+0.05*dir(-25), dir(-25));[/asy]](http://latex.artofproblemsolving.com/9/8/9/9891c6f80eef7810f1c8cebaf38d13ccfa8607b9.png)
Solution
![[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C); draw(C--A--D^^B--O^^Circle(O,1)); pair point=O; draw(C--D); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$O$", O, dir(305)); label("$5$", B--O, dir(O--B)*dir(90)); label("$60^\circ$", dir(185), dir(185)); label("$60^\circ$", B+0.05*dir(-25), dir(-25));[/asy]](http://latex.artofproblemsolving.com/0/6/3/063f3aa056d04cd23ddf7d49272b0eedc273edab.png)
Since 
is inscribed and intersects an arc of length 
, 
. Thus, 
is a 
right triangle. Thus, 
and 
. Since 
and 
are both radii, 
and 
. Since 
is inscribed in a semicircle, it's a right angle, and 
is also a right triangle. Thus, 
and 
. Finally, 
.
See Also
| 1985 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 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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