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

Latest revision as of 00:38, 20 August 2024

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 1

Since $\overline{AB}$ is tangent to the circle, $\triangle OAB$ is a right triangle. This means that $OA = 1$ , $AB = \tan \theta$ 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 $\boxed{\textbf{(D)} \dfrac{1}{1+\sin \theta}}$ .

Solution 2

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 3 (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:

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

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

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

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

Solution 4

Let $OC$ = x, $OB$ = h, and $AB$ = y. $AC$ = $OA$ - $OC$ .

Because $OC$ = x, and $OA$ = 1 (given in the problem), $AC$ = 1-x.

Using the Angle Bisector Theorem, $\frac{h}{y}$ = $\frac{x}{1-x}$ $\Longrightarrow$ h(1-x) = xy. Solving for x gives us x = $\frac{h}{h+y}$ .

$\sin\theta = \frac{opposite}{hypotenuse} = \frac{y}{h}$ . Solving for y gives us y = $h \sin\theta$ .

Substituting this for y in our initial equation yields x = $\dfrac{h}{h+h\sin \theta}$ .

Using the distributive property, x = $\dfrac{h}{h(1+\sin \theta)}$ and finally $\dfrac{1}{1+\sin \theta}$ or $\boxed{\textbf{(D)}}$

Solution 5

Since $\overline{AB}$ is tangent to the circle, $\angle OAB=90^{\circ}$ and thus we can use trig ratios directly.

$\sin{\theta}=\frac{\overline{AB}}{\overline{BO}}, \cos{\theta}=\frac{1}{\overline{BO}}, \tan{\theta}=\overline{AB}$

By the angle bisector theorem, we have

$\frac{\overline{OB}}{\overline{AB}}=\frac{\overline{OC}}{\overline{CA}}$

Seeing the resemblance of the ratio on the left-hand side to $\sin{\theta},$ we turn the ratio around to allow us to plug in $\sin{\theta}.$ Another source of motivation for this also lies in the idea of somehow adding 1 to the right-hand side so that we can substitute for a given value, i.e. $\overline{OA}=1$ , and flipping the fraction will preserve the $\overline{OC}$ , whilst adding one right now would make the equation remain in direct terms of $\overline{CA}.$

$\frac{\overline{AB}}{\overline{OB}}=\sin{\theta}=\frac{\overline{CA}}{\overline{OC}}\Rightarrow \sin{\theta}+1=\frac{\overline{CA}+\overline{OC}}{\overline{OC}}=\frac{1}{\overline{OC}}$

$\sin{\theta}+1=\frac{1}{\overline{OC}} \Rightarrow \boxed{\overline{OC}=\frac{1}{\sin{\theta}+1}}$

Solution 6 (tangent half angle)

$\angle CBO = 45^{\circ} - \frac{\theta}{2}, \angle ACB = 45^{\circ} + \frac{\theta}{2}, OB = \frac{1}{\cos(\theta)}$ . By sine law, $\frac{OC}{\sin(\angle CBO)} = \frac{OB}{\sin(\angle OCB)} = \frac{OB}{\sin(\angle ACB)}$

$OC = \frac{\sin(45^{\circ} - \frac{\theta}{2})}{\sin(45^{\circ} + \frac{\theta}{2})}OB = \frac{\sin(45^{\circ} - \frac{\theta}{2})}{\cos(45^{\circ} - \frac{\theta}{2})}OB = \tan(45^{\circ} - \frac{\theta}{2})OB = \frac{1-\tan(\theta/2)}{1+\tan(\theta/2)}OB$

Let $t = \tan(\theta/2)$ . $OC = \frac{1-t}{1+t}OB = \frac{1-t^2}{1+2t+t^2}$ . Because $\sin(\theta) = \frac{2t}{1+t^2}$ and $\cos(\theta) = \frac{1-t^2}{1+t^2}$ , $OC = \frac{\cos(\theta)}{1+\sin(\theta)}OB = \boxed{\textbf{(D)} \dfrac{1}{1+\sin \theta}}$

Solution 7 (if you forgot angle bisector but remember LoS)

Let $x=\overline{OC}$ , and let $\angle OBC=\angle ABC=\alpha$ . We know that $\overline{AC}=\overline{OA}-\overline{OC}=1-x$ . By the Law of Sines, $\dfrac{\sin\alpha}x=\dfrac{\sin\theta}{BC}$

and

$\dfrac{\sin\alpha}{1-x}=\dfrac{\sin90^\circ}{BC}=\dfrac1{BC}.$

Combining the two give $\dfrac{\sin\alpha}x=\sin\theta\cdot\dfrac{\sin\alpha}{1-x}$ .

Solving, this gives $\boxed{x=\frac{1}{\sin{\theta}+1}}$ .

~Technodoggo

Video Solution

https://youtu.be/zGnRV6eNiV0

@@ Line 29: / Line 29: @@
 <math>\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}</math>
-== Solution ==
+== Solution 1 ==
-Since <math>\overline{AB}</math> is tangent to the circle, <math>\triangle OAB</math> is a right triangle. This means that <math>OA = 1</math>, <math>BA = \tan \theta</math> and <math>OB = \sec \theta</math>. By the [[Angle Bisector Theorem]], <cmath> \frac{OB}{OC} = \frac{AB}{AC} \Longrightarrow AC \sec \theta = OC \tan \theta </cmath> We multiply both sides by <math>\cos \theta</math> to simplify the trigonometric functions, <cmath> AC=OC \sin \theta </cmath> Since <math>AC + OC = 1</math>, <math>1 - OC = OC \sin \theta \Longrightarrow</math> <math>OC = \dfrac{1}{1+\sin \theta}</math>. Therefore, the answer is <math>\boxed{\textbf{(D)} \dfrac{1}{1+\sin \theta}}</math>.
+Since <math>\overline{AB}</math> is tangent to the circle, <math>\triangle OAB</math> is a right triangle. This means that <math>OA = 1</math>, <math>AB = \tan \theta</math> and <math>OB = \sec \theta</math>. By the [[Angle Bisector Theorem]], <cmath> \frac{OB}{OC} = \frac{AB}{AC} \Longrightarrow AC \sec \theta = OC \tan \theta </cmath> We multiply both sides by <math>\cos \theta</math> to simplify the trigonometric functions, <cmath> AC=OC \sin \theta </cmath> Since <math>AC + OC = 1</math>, <math>1 - OC = OC \sin \theta \Longrightarrow</math> <math>OC = \dfrac{1}{1+\sin \theta}</math>. Therefore, the answer is <math>\boxed{\textbf{(D)} \dfrac{1}{1+\sin \theta}}</math>.
+== Solution 2 ==
+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 3 (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>
+<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{\text{Opposite}}{\text{Hypotenuse}}</math>
+<math>\cos\theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}</math>
+<math>\tan\theta =\frac{\text{Opposite}}{\text{Adjacent}}</math>.
+With a bit of guess and check, we get that the answer is <math>\boxed{D}</math>.
+== Solution 4 ==
+Let <math>OC</math> = x, <math>OB</math> = h, and <math>AB</math> = y. <math>AC</math> = <math>OA</math> - <math>OC</math>.
+Because <math>OC</math> = x, and <math>OA</math> = 1 (given in the problem), <math>AC</math> = 1-x.
+Using the [[Angle Bisector Theorem]], <math>\frac{h}{y}</math> = <math>\frac{x}{1-x}</math> <math>\Longrightarrow</math> h(1-x) = xy. Solving for x gives us x = <math>\frac{h}{h+y}</math>.
+<math>\sin\theta = \frac{opposite}{hypotenuse} = \frac{y}{h}</math>. Solving for y gives us y = <math>h \sin\theta</math>.
+Substituting this for y in our initial equation yields x = <math>\dfrac{h}{h+h\sin \theta}</math>.
+Using the distributive property, x = <math>\dfrac{h}{h(1+\sin \theta)}</math> and finally <math>\dfrac{1}{1+\sin \theta}</math> or <math>\boxed{\textbf{(D)}}</math>
+== Solution 5 ==
+Since <math>\overline{AB}</math> is tangent to the circle, <math>\angle OAB=90^{\circ}</math> and thus we can use trig ratios directly.
+<cmath>\sin{\theta}=\frac{\overline{AB}}{\overline{BO}}, \cos{\theta}=\frac{1}{\overline{BO}}, \tan{\theta}=\overline{AB}</cmath>
+By the angle bisector theorem, we have
+<cmath>\frac{\overline{OB}}{\overline{AB}}=\frac{\overline{OC}}{\overline{CA}}</cmath>
+Seeing the resemblance of the ratio on the left-hand side to <math>\sin{\theta},</math> we turn the ratio around to allow us to plug in <math>\sin{\theta}.</math> Another source of motivation for this also lies in the idea of somehow adding 1 to the right-hand side so that we can substitute for a given value, i.e. <math>\overline{OA}=1</math>, and flipping the fraction will preserve the <math>\overline{OC}</math>, whilst adding one right now would make the equation remain in direct terms of <math>\overline{CA}.</math>
+<cmath>\frac{\overline{AB}}{\overline{OB}}=\sin{\theta}=\frac{\overline{CA}}{\overline{OC}}\Rightarrow \sin{\theta}+1=\frac{\overline{CA}+\overline{OC}}{\overline{OC}}=\frac{1}{\overline{OC}}</cmath>
+<cmath>\sin{\theta}+1=\frac{1}{\overline{OC}} \Rightarrow \boxed{\overline{OC}=\frac{1}{\sin{\theta}+1}}</cmath>
+== Solution 6 (tangent half angle) ==
+<math>\angle CBO = 45^{\circ} - \frac{\theta}{2}, \angle ACB = 45^{\circ} + \frac{\theta}{2}, OB = \frac{1}{\cos(\theta)}</math>.
+By sine law, <math>\frac{OC}{\sin(\angle CBO)} = \frac{OB}{\sin(\angle OCB)} = \frac{OB}{\sin(\angle ACB)}</math>
+<cmath>OC = \frac{\sin(45^{\circ} - \frac{\theta}{2})}{\sin(45^{\circ} + \frac{\theta}{2})}OB = \frac{\sin(45^{\circ} - \frac{\theta}{2})}{\cos(45^{\circ} - \frac{\theta}{2})}OB = \tan(45^{\circ} - \frac{\theta}{2})OB = \frac{1-\tan(\theta/2)}{1+\tan(\theta/2)}OB</cmath>
+Let <math>t = \tan(\theta/2)</math>. <math>OC = \frac{1-t}{1+t}OB = \frac{1-t^2}{1+2t+t^2}</math>. Because <math>\sin(\theta) = \frac{2t}{1+t^2}</math> and <math>\cos(\theta) = \frac{1-t^2}{1+t^2}</math>, <cmath>OC = \frac{\cos(\theta)}{1+\sin(\theta)}OB = \boxed{\textbf{(D)} \dfrac{1}{1+\sin \theta}}</cmath>
+==Solution 7 (if you forgot angle bisector but remember LoS)==
+Let <math>x=\overline{OC}</math>, and let <math>\angle OBC=\angle ABC=\alpha</math>. We know that <math>\overline{AC}=\overline{OA}-\overline{OC}=1-x</math>. By the Law of Sines,
+<cmath>\dfrac{\sin\alpha}x=\dfrac{\sin\theta}{BC}</cmath>
+and
+<cmath>\dfrac{\sin\alpha}{1-x}=\dfrac{\sin90^\circ}{BC}=\dfrac1{BC}.</cmath>
+Combining the two give <math>\dfrac{\sin\alpha}x=\sin\theta\cdot\dfrac{\sin\alpha}{1-x}</math>.
+Solving, this gives <math>\boxed{x=\frac{1}{\sin{\theta}+1}}</math>.
+~Technodoggo
+== Video Solution ==
+https://youtu.be/zGnRV6eNiV0
 == 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
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