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Difference between revisions of "2000 AMC 12 Problems/Problem 17"

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Alternatively, one could notice that OC approaches the value 1/2 as theta gets close to 90 degrees.  The only choice that is consistent with this is (D).
 
Alternatively, one could notice that OC approaches the value 1/2 as theta gets close to 90 degrees.  The only choice that is consistent with this is (D).
 +
 +
==Solution (with minimal trig)==
 +
Let's assign a value to <math>\theta</math> so we don't have to use trig functions to solve.  <math>60</math> is a good value for <math>\theta</math>, because then we have a <math>30-60-90 \triangle</math> -- <math>\angle BAC=90</math> because <math>AB</math> is tangent to Circle <math>O</math>.
 +
 +
Using our special right triangle, since <math>AO=1</math>, <math>OB=2</math>, and <math>AB=\sqrt{3}</math>.
 +
 +
Let <math>OC=x</math>.  Then <math>CA=1-x</math>.  since <math>BC</math> bisects <math>\angle ABO</math>, we can use the angle bisector theorem:
 +
 +
<math>\frac{2}{x}=\frac{\sqrt{3}}{1-x}</math>
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 +
<math>2-2x=\sqrt{3}x</math>
 +
 +
<math>2=(\sqrt{3}+2)x</math>
 +
 +
<math>x=\frac{2}{\sqrt{3}+2}</math>.
 +
 +
Now, we only have to use a bit of trig to guess and check: the only trig facts we need to know to finish the problem is:
 +
 +
<math>sin\theta =\frac{Opposite}{Hypotenuse}</math>
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 +
<math>cos\theta =\frac{Adjacent}{Hypotenuse}</math>
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 +
<math>tan\theta =\frac{Opposite}{Adjacent}</math>.
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 +
With a bit of guess and check, we get that the answer is <math>\boxed{D}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 11:06, 24 March 2016

Problem

A circle centered at $O$ has radius $1$ and contains the point $A$. The segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, then $OC =$

[asy] import olympiad; size(6cm); unitsize(1cm); defaultpen(fontsize(8pt)+linewidth(.8pt)); labelmargin=0.2; dotfactor=3; pair O=(0,0); pair A=(1,0); pair B=(1,1.5); pair D=bisectorpoint(A,B,O); pair C=extension(B,D,O,A); draw(Circle(O,1)); draw(O--A--B--cycle); draw(B--C); label("$O$",O,SW); dot(O); label("$\theta$",(0.1,0.05),ENE); dot(C); label("$C$",C,S); dot(A); label("$A$",A,E); dot(B); label("$B$",B,E);[/asy]

$\text {(A)}\ \sec^2 \theta - \tan \theta \qquad \text {(B)}\ \frac 12 \qquad \text {(C)}\ \frac{\cos^2 \theta}{1 + \sin \theta}\qquad \text {(D)}\ \frac{1}{1+\sin\theta} \qquad \text {(E)}\ \frac{\sin \theta}{\cos^2 \theta}$

Solution

Since $\overline{AB}$ is tangent to the circle, $\triangle OAB$ is a right triangle. This means that $OA = 1$, $BA = \tan \theta$ and $OB = \sec \theta$. By the Angle Bisector Theorem, \[\frac{OB}{OC} = \frac{AB}{AC} \Longrightarrow AC \sec \theta = OC \tan \theta\] We multiply both sides by $\cos \theta$ to simplify the trigonometric functions, \[AC=OC \sin \theta\] Since $AC + OC = 1$, $1 - OC = OC \sin \theta \Longrightarrow$ $OC = \dfrac{1}{1+\sin \theta}$. Therefore, the answer is $\boxed{\textbf{(D)} \dfrac{1}{1+\sin \theta}}$.


Alternatively, one could notice that OC approaches the value 1/2 as theta gets close to 90 degrees. The only choice that is consistent with this is (D).

Solution (with minimal trig)

Let's assign a value to $\theta$ so we don't have to use trig functions to solve. $60$ is a good value for $\theta$, because then we have a $30-60-90 \triangle$ -- $\angle BAC=90$ because $AB$ is tangent to Circle $O$.

Using our special right triangle, since $AO=1$, $OB=2$, and $AB=\sqrt{3}$.

Let $OC=x$. Then $CA=1-x$. since $BC$ bisects $\angle ABO$, we can use the angle bisector theorem:

$\frac{2}{x}=\frac{\sqrt{3}}{1-x}$

$2-2x=\sqrt{3}x$

$2=(\sqrt{3}+2)x$

$x=\frac{2}{\sqrt{3}+2}$.

Now, we only have to use a bit of trig to guess and check: the only trig facts we need to know to finish the problem is:

$sin\theta =\frac{Opposite}{Hypotenuse}$

$cos\theta =\frac{Adjacent}{Hypotenuse}$

$tan\theta =\frac{Opposite}{Adjacent}$.

With a bit of guess and check, we get that the answer is $\boxed{D}$.

See also

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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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