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Difference between revisions of "2020 AMC 12B Problems/Problem 10"
m (→‎Video Solution)
 
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==Problem==
 
==Problem==
−
 
 
In unit square <math>ABCD,</math> the inscribed circle <math>\omega</math> intersects <math>\overline{CD}</math> at <math>M,</math> and <math>\overline{AM}</math> intersects <math>\omega</math> at a point <math>P</math> different from <math>M.</math> What is <math>AP?</math>
 
In unit square <math>ABCD,</math> the inscribed circle <math>\omega</math> intersects <math>\overline{CD}</math> at <math>M,</math> and <math>\overline{AM}</math> intersects <math>\omega</math> at a point <math>P</math> different from <math>M.</math> What is <math>AP?</math>
  
 
<math>\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}</math>
 
<math>\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}</math>
  
−
==Solution 1 (Angle Chasing/Trig)==
+
==Diagram==
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(180);
 +
pair A, B, C, D, M, O, P;
 +
O = origin;
 +
A = (-1/2,-1/2);
 +
B = (-1/2,1/2);
 +
C = (1/2,1/2);
 +
D = (1/2,-1/2);
 +
M = midpoint(C--D);
 +
path p;
 +
p = Circle(O,1/2);
 +
P = intersectionpoints(A--M,p)[0];
 +
dot("$\omega$",O,1.5*N,linewidth(4));
 +
dot("$A$",A,1.5*SW,linewidth(4));
 +
dot("$B$",B,1.5*NW,linewidth(4));
 +
dot("$C$",C,1.5*NE,linewidth(4));
 +
dot("$D$",D,1.5*SE,linewidth(4));
 +
dot("$M$",M,1.5*E,linewidth(4));
 +
dot("$P$",P,1.5*N,linewidth(4));
 +
draw(A--B--C--D--cycle^^A--M^^p);
 +
</asy>
 +
~MRENTHUSIASM
 +
 
 +
==Solution 1 (Similar Triangles)==
 +
Call the midpoint of <math>\overline{AB}</math> point <math>N.</math> Draw in <math>\overline{NM}</math> and <math>\overline{NP}.</math> Note that <math>\angle{NPM}=90^{\circ}</math> due to Thales's Theorem.
 +
<asy>
 +
/* Made by QIDb602; edited by MRENTHUSIASM */
 +
size(180);
 +
pair A, B, C, D, M, N, O, P;
 +
O = origin;
 +
A = (-1/2,-1/2);
 +
B = (-1/2,1/2);
 +
C = (1/2,1/2);
 +
D = (1/2,-1/2);
 +
M = midpoint(C--D);
 +
N = midpoint(A--B);
 +
path p;
 +
p = Circle(O,1/2);
 +
P = intersectionpoints(A--M,p)[0];
 +
fill(N--A--M--cycle,yellow);
 +
dot("$\omega$",O,1.5*(0,1),linewidth(4));
 +
dot("$A$",A,1.5*SW,linewidth(4));
 +
dot("$B$",B,1.5*NW,linewidth(4));
 +
dot("$C$",C,1.5*NE,linewidth(4));
 +
dot("$D$",D,1.5*SE,linewidth(4));
 +
dot("$M$",M,1.5*E,linewidth(4));
 +
dot("$N$",N,1.5*W,linewidth(4));
 +
dot("$P$",P,1.5*dir(60),linewidth(4));
 +
markscalefactor=0.00625;
 +
draw(rightanglemark(A,N,M),red);
 +
draw(rightanglemark(N,P,A),red);
 +
draw(A--B--C--D--cycle^^A--M^^P--N--M^^p);
 +
</asy>
 +
Using the Pythagorean theorem, <math>AM=\frac{\sqrt{5}}{2}.</math> Now we just need to find <math>AP</math> using similar triangles <math>\triangle APN\sim\triangle ANM:</math>
 +
<cmath>\begin{align*}
 +
\frac{AP}{AN}&=\frac{AN}{AM} \\
 +
\frac{AP}{1/2}&=\frac{1/2}{\sqrt5/2} \\
 +
AP&=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.
 +
\end{align*}</cmath>
 +
~QIDb602
 +
 
 +
==Solution 2 (Inscribed Angle Theorem and Pythagorean Theorem)==
 +
Let <math>N</math> be the midpoint of <math>\overline{AB},</math> from which <math>\angle ANM=90^\circ.</math> Note that <math>\angle NPM=90^\circ</math> by the Inscribed Angle Theorem.
 +
 
 +
We have the following diagram:
 +
<asy>
 +
/* Made by MRENTHUSIASM */
 +
size(180);
 +
pair A, B, C, D, M, N, O, P;
 +
O = origin;
 +
A = (-1/2,-1/2);
 +
B = (-1/2,1/2);
 +
C = (1/2,1/2);
 +
D = (1/2,-1/2);
 +
M = midpoint(C--D);
 +
N = midpoint(A--B);
 +
path p;
 +
p = Circle(O,1/2);
 +
P = intersectionpoints(A--M,p)[0];
 +
fill(N--P--A--cycle,yellow);
 +
fill(N--P--M--cycle,green);
 +
dot("$\omega$",O,1.5*(0,1),linewidth(4));
 +
dot("$A$",A,1.5*SW,linewidth(4));
 +
dot("$B$",B,1.5*NW,linewidth(4));
 +
dot("$C$",C,1.5*NE,linewidth(4));
 +
dot("$D$",D,1.5*SE,linewidth(4));
 +
dot("$M$",M,1.5*E,linewidth(4));
 +
dot("$N$",N,1.5*W,linewidth(4));
 +
dot("$P$",P,1.5*dir(60),linewidth(4));
 +
markscalefactor=0.00625;
 +
draw(rightanglemark(A,N,M),red);
 +
draw(rightanglemark(N,P,A),red);
 +
draw(A--B--C--D--cycle^^A--M^^P--N--M^^p);
 +
</asy>
 +
Since <math>AN=\frac12</math> and <math>NM=1,</math> we get <math>AM=\frac{\sqrt5}{2}</math> by the Pythagorean Theorem.
 +
 
 +
Let <math>AP=x.</math> It follows that <math>PM=\frac{\sqrt5}{2}-x.</math> Applying the Pythagorean Theorem to right <math>\triangle ANP</math> gives <math>NP^2=\left(\frac12\right)^2-x^2,</math> and applying the Pythagorean Theorem to right <math>\triangle MNP</math> gives <math>NP^2=1^2-\left(\frac{\sqrt5}{2}-x\right)^2.</math> Equating the expressions for <math>NP^2</math> produces
 +
<cmath>\begin{align*}
 +
\left(\frac12\right)^2-x^2&=1^2-\left(\frac{\sqrt5}{2}-x\right)^2 \\
 +
\frac14-x^2&=1-\frac54+\sqrt5x-x^2 \\
 +
\frac12&=\sqrt5x.
 +
\end{align*}</cmath>
 +
Finally, dividing both sides by <math>\sqrt5</math> and then rationalizing the denominator, we obtain <math>x=\frac{1}{2\sqrt5}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.</math>
 +
 
 +
~MRENTHUSIASM
 +
 
 +
==Solution 3 (Power of a Point)==
 +
Let circle <math>\omega</math> intersect <math>\overline{AB}</math> at point <math>N</math>. By Power of a Point, we have <math>AN^2=AP\cdot AM</math>. We know <math>AN=\frac{1}{2}</math> because <math>N</math> is the midpoint of <math>\overline{AB}</math>, and we can easily find <math>AM</math> by the Pythagorean Theorem, which gives us <math>AM=\sqrt{1^2+\left(\frac{1}{2}\right)^2}=\frac{\sqrt{5}}{2}</math>. Our equation is now <math>\frac{1}{4}=AP\cdot \frac{\sqrt{5}}{2}</math>, or <math>AP=\frac{2}{4\sqrt{5}}=\frac{1}{2\sqrt{5}}=\frac{\sqrt{5}}{2\cdot 5}</math>, thus our answer is
 +
<math>\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.</math>
 +
 
 +
~Argonauts16
 +
 
 +
==Solution 4 (Trigonometry)==
 
Let <math>O</math> be the center of the circle and the point of tangency between <math>\omega</math> and <math>\overline{AD}</math> be represented by <math>K</math>. We know that <math>\overline{AK} = \overline{KD} = \overline{DM} = \frac{1}{2}</math>. Consider the right triangle <math>\bigtriangleup ADM</math>. Let <math>\measuredangle AMD = \theta</math>.  
 
Let <math>O</math> be the center of the circle and the point of tangency between <math>\omega</math> and <math>\overline{AD}</math> be represented by <math>K</math>. We know that <math>\overline{AK} = \overline{KD} = \overline{DM} = \frac{1}{2}</math>. Consider the right triangle <math>\bigtriangleup ADM</math>. Let <math>\measuredangle AMD = \theta</math>.  
  
Line 12: Line 125:
 
We can now use Law of Cosines on <math>\angle{POM}</math> to find the length of <math>{PM}</math> and subtract it from the length of <math>{AM}</math> to find <math>{AP}</math>. Since <math>\cos{\theta} = \frac{1}{\sqrt{5}}</math> and <math>\sin{\theta} = \frac{2}{\sqrt{5}}</math>, the double angle formula tells us that <math>\cos{2\theta} = -\frac{3}{5}</math>. We have
 
We can now use Law of Cosines on <math>\angle{POM}</math> to find the length of <math>{PM}</math> and subtract it from the length of <math>{AM}</math> to find <math>{AP}</math>. Since <math>\cos{\theta} = \frac{1}{\sqrt{5}}</math> and <math>\sin{\theta} = \frac{2}{\sqrt{5}}</math>, the double angle formula tells us that <math>\cos{2\theta} = -\frac{3}{5}</math>. We have
 
<cmath>
 
<cmath>
−
PM^2 = \frac{1}{2} - \frac{1}{2}\cos{2\theta} \implies PM = \frac{2\sqrt{5}}{5}
+
PM^2 = \frac{1}{2} - \frac{1}{2}\cos{2\theta} \implies PM = \frac{2\sqrt{5}}{5}.
 
</cmath>
 
</cmath>
−
By Pythagorean theorem, we find that <math>AM = \frac{\sqrt{5}}{2} \implies \boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>
+
By Pythagorean theorem, we find that <math>AM = \frac{\sqrt{5}}{2} \implies AP=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
  
 
~awesome1st
 
~awesome1st
−
==Solution 2(Coordinate Bash)==
+
 
 +
==Solution 5 (Trigonometry)==
 +
Take <math>O</math> as the center and draw segment <math>ON</math> perpendicular to <math>AM</math>, <math>ON\cap AM=N</math>, link <math>OM</math>. Then we have <math>OM\parallel AD</math>. So <math>\angle DAM=\angle OMA</math>. Since <math>AD=2AM=2OM=1</math>, we have <math>\cos\angle DAM=\cos\angle OMP=\frac{2}{\sqrt{5}}</math>. As a result, <math>NM=OM\cos\angle OMP=\frac{1}{2}\cdot \frac{2}{\sqrt{5}}=\frac{1}{\sqrt{5}}.</math> Thus <math>PM=2NM=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}</math>. Since <math>AM=\frac{\sqrt{5}}{2}</math>, we have <math>AP=AM-PM=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
 +
 
 +
~FANYUCHEN20020715
 +
 
 +
==Solution 6 (Coordinate Geometry)==
  
 
Place circle <math>\omega</math> in the Cartesian plane such that the center lies on the origin. Then we can easily find the equation for <math>\omega</math> as <math>x^2+y^2=\frac{1}{4}</math>, because it is not translated and the radius is <math>\frac{1}{2}</math>.  
 
Place circle <math>\omega</math> in the Cartesian plane such that the center lies on the origin. Then we can easily find the equation for <math>\omega</math> as <math>x^2+y^2=\frac{1}{4}</math>, because it is not translated and the radius is <math>\frac{1}{2}</math>.  
Line 25: Line 144:
 
We substitute this into the equation for the circle to get <math>x^2+\left(-2x-\frac{1}{2}\right)^2=\frac{1}{4}</math>, or <math>x^2+4x^2+2x+\frac{1}{4}=\frac{1}{4}</math>. Simplifying gives <math>x(5x+2)=0</math>. The roots of this quadratic are <math>x=0</math> and <math>x=-\frac{2}{5}</math>, but if <math>x=0</math> we get point <math>M</math>, so we only want <math>x=-\frac{2}{5}</math>.
 
We substitute this into the equation for the circle to get <math>x^2+\left(-2x-\frac{1}{2}\right)^2=\frac{1}{4}</math>, or <math>x^2+4x^2+2x+\frac{1}{4}=\frac{1}{4}</math>. Simplifying gives <math>x(5x+2)=0</math>. The roots of this quadratic are <math>x=0</math> and <math>x=-\frac{2}{5}</math>, but if <math>x=0</math> we get point <math>M</math>, so we only want <math>x=-\frac{2}{5}</math>.
  
−
We plug this back into the linear equation to find <math>y=\frac{3}{10}</math>, and so <math>P=\left(-\frac{2}{5}, \frac{3}{10}\right)</math>. Finally, we use distance formula on <math>A</math> and <math>P</math> to get <math>AP=\sqrt{\left(-\frac{5}{10}+\frac{4}{10}\right)^2+\left(\frac{5}{10}-\frac{3}{10}\right)^2}=\sqrt{\frac{1}{100}+\frac{4}{100}}=\boxed{\mathbf{(B) } \frac{\sqrt{5}}{10}}</math>.
+
We plug this back into the linear equation to find <math>y=\frac{3}{10}</math>, and so <math>P=\left(-\frac{2}{5}, \frac{3}{10}\right)</math>. Finally, we use distance formula on <math>A</math> and <math>P</math> to get <math>AP=\sqrt{\left(-\frac{5}{10}+\frac{4}{10}\right)^2+\left(\frac{5}{10}-\frac{3}{10}\right)^2}=\sqrt{\frac{1}{100}+\frac{4}{100}}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}</math>.
  
 
~Argonauts16
 
~Argonauts16
  
−
==Solution 3(Power of a Point)==
+
==Video Solution (HOW TO THINK CREATIVELY!!!)==
 +
https://youtu.be/gd6VvCKMdu4
 +
 
 +
~Education, the Study of Everything
  
−
Let circle <math>\omega</math> intersect <math>\overline{AB}</math> at point <math>N</math>. By Power of a Point, we have <math>AN^2=AP\cdot AM</math>. We know <math>AN=\frac{1}{2}</math> because <math>N</math> is the midpoint of <math>\overline{AB}</math>, and we can easily find <math>AM</math> by the Pythagorean Theorem, which gives us <math>AM=\sqrt{1^2+\left(\frac{1}{2}\right)^2}=\frac{\sqrt{5}}{2}</math>. Our equation is now <math>\frac{1}{4}=AP\cdot \frac{\sqrt{5}}{2}</math>, or <math>AP=\frac{2}{4\sqrt{5}}=\frac{1}{2\sqrt{5}}=\frac{\sqrt{5}}{2\cdot 5}</math>, thus our answer is
 
−
<math>\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.</math>
 
  
−
~Argonauts16
 
  
−
==Solution 4==
 
−
Take <math>O</math> as the center and draw segment <math>ON</math> perpendicular to <math>AM</math>, <math>ON\cap AM=N</math>, link <math>OM</math>. Then we have <math>OM\parallel AD</math>. So <math>\angle DAM=\angle OMA</math>. Since <math>AD=2AM=2OM=1</math>, we have <math>\cos\angle DAM=\cos\angle OMP=\frac{2}{\sqrt{5}}</math>. As a result, <math>NM=OM\cos\angle OMP=\frac{1}{2}\cdot \frac{2}{\sqrt{5}}=\frac{1}{\sqrt{5}}.</math> Thus <math>PM=2NM=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}</math>. Since <math>AM=\frac{\sqrt{5}}{2}</math>, we have <math>AP=AM-PM=\frac{\sqrt{5}}{10}</math>. Put <math>\boxed{B}</math>.
 
  
−
~FANYUCHEN20020715
 
  
−
==Solution 5 (Similar Triangles)==
 
−
Call the midpoint of <math>\overline{AB}</math> point <math>N</math>. Draw in <math>\overline{NM}</math> and <math>\overline{NP}</math>. Note that <math>\angle{NPM}=90^{\circ}</math> due to Thales's Theorem.
 
  
−
<asy>
 
−
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
 
−
draw(circle((0.5,0.5),0.5));
 
−
draw((0,0)--(0,0.5)--(1,0.5)--cycle);
 
−
label("A",(0,0),SW);
 
−
label("B",(0,1),NW);
 
−
label("C",(1,1),NE);
 
−
label("D",(1,0),SE);
 
−
label("M",(1,0.5),E);
 
−
label("P",(0.2,0.1),S);
 
−
label("N",(0,0.5),W);
 
−
draw((0,0.5)--(0.2,0.1));
 
−
markscalefactor=0.007;
 
−
draw(rightanglemark((0,0.5),(0.2,0.1),(1,0.5)));
 
−
</asy>
 
−
Using the Pythagorean theorem, <math>AM=\frac{\sqrt{5}}{2}</math>. Now we just need to find <math>AP</math> using similar triangles.
 
−
<cmath>\triangle APN\sim\triangle ANM\Rightarrow\frac{AP}{AN}=\frac{AN}{AM}\Rightarrow\frac{AP}{\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{\sqrt{5}}{2}}\Rightarrow AP=\boxed{\textbf{(B) }\frac{\sqrt{5}}{10}}</cmath>
 
−
~QIDb602
 
  
−
==Video Solution==
+
==Video Solution by TheBeautyOfMath==
 
https://youtu.be/6ujfjGLzVoE
 
https://youtu.be/6ujfjGLzVoE
−
 
−
~IceMatrix
 
−
 
−
==Solution 6 intersecting chords==
 
−
Label the midpoint of <math>AD</math> as <math>N</math>, and let <math>Q</math> be the intersection of <math>ON</math> and <math>AM</math>
 
−
Then <math>MQ=AQ=\frac{1}{2}AM = \frac{\sqrt{5}}{4}</math> and <math>NQ=\frac{1}{4}</math>
 
−
 
−
Then <math>\frac{1}{4}*\frac{3}{4}=PQ*QM</math> and <math>AP=AQ-PQ</math>
 
−
 
−
~ERMSCoach
 
  
 
==See Also==
 
==See Also==

Latest revision as of 15:14, 11 September 2023

Problem

In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$

$\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$

Diagram

[asy] /* Made by MRENTHUSIASM */ size(180); pair A, B, C, D, M, O, P; O = origin; A = (-1/2,-1/2); B = (-1/2,1/2); C = (1/2,1/2); D = (1/2,-1/2); M = midpoint(C--D); path p; p = Circle(O,1/2); P = intersectionpoints(A--M,p)[0]; dot("$\omega$",O,1.5*N,linewidth(4)); dot("$A$",A,1.5*SW,linewidth(4)); dot("$B$",B,1.5*NW,linewidth(4)); dot("$C$",C,1.5*NE,linewidth(4)); dot("$D$",D,1.5*SE,linewidth(4)); dot("$M$",M,1.5*E,linewidth(4)); dot("$P$",P,1.5*N,linewidth(4)); draw(A--B--C--D--cycle^^A--M^^p); [/asy] ~MRENTHUSIASM

Solution 1 (Similar Triangles)

Call the midpoint of $\overline{AB}$ point $N.$ Draw in $\overline{NM}$ and $\overline{NP}.$ Note that $\angle{NPM}=90^{\circ}$ due to Thales's Theorem. [asy] /* Made by QIDb602; edited by MRENTHUSIASM */ size(180); pair A, B, C, D, M, N, O, P; O = origin; A = (-1/2,-1/2); B = (-1/2,1/2); C = (1/2,1/2); D = (1/2,-1/2); M = midpoint(C--D); N = midpoint(A--B); path p; p = Circle(O,1/2); P = intersectionpoints(A--M,p)[0]; fill(N--A--M--cycle,yellow); dot("$\omega$",O,1.5*(0,1),linewidth(4)); dot("$A$",A,1.5*SW,linewidth(4)); dot("$B$",B,1.5*NW,linewidth(4)); dot("$C$",C,1.5*NE,linewidth(4)); dot("$D$",D,1.5*SE,linewidth(4)); dot("$M$",M,1.5*E,linewidth(4)); dot("$N$",N,1.5*W,linewidth(4)); dot("$P$",P,1.5*dir(60),linewidth(4)); markscalefactor=0.00625; draw(rightanglemark(A,N,M),red); draw(rightanglemark(N,P,A),red); draw(A--B--C--D--cycle^^A--M^^P--N--M^^p); [/asy] Using the Pythagorean theorem, $AM=\frac{\sqrt{5}}{2}.$ Now we just need to find $AP$ using similar triangles $\triangle APN\sim\triangle ANM:$ \begin{align*} \frac{AP}{AN}&=\frac{AN}{AM} \\ \frac{AP}{1/2}&=\frac{1/2}{\sqrt5/2} \\ AP&=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}. \end{align*} ~QIDb602

Solution 2 (Inscribed Angle Theorem and Pythagorean Theorem)

Let $N$ be the midpoint of $\overline{AB},$ from which $\angle ANM=90^\circ.$ Note that $\angle NPM=90^\circ$ by the Inscribed Angle Theorem.

We have the following diagram: [asy] /* Made by MRENTHUSIASM */ size(180); pair A, B, C, D, M, N, O, P; O = origin; A = (-1/2,-1/2); B = (-1/2,1/2); C = (1/2,1/2); D = (1/2,-1/2); M = midpoint(C--D); N = midpoint(A--B); path p; p = Circle(O,1/2); P = intersectionpoints(A--M,p)[0]; fill(N--P--A--cycle,yellow); fill(N--P--M--cycle,green); dot("$\omega$",O,1.5*(0,1),linewidth(4)); dot("$A$",A,1.5*SW,linewidth(4)); dot("$B$",B,1.5*NW,linewidth(4)); dot("$C$",C,1.5*NE,linewidth(4)); dot("$D$",D,1.5*SE,linewidth(4)); dot("$M$",M,1.5*E,linewidth(4)); dot("$N$",N,1.5*W,linewidth(4)); dot("$P$",P,1.5*dir(60),linewidth(4)); markscalefactor=0.00625; draw(rightanglemark(A,N,M),red); draw(rightanglemark(N,P,A),red); draw(A--B--C--D--cycle^^A--M^^P--N--M^^p); [/asy] Since $AN=\frac12$ and $NM=1,$ we get $AM=\frac{\sqrt5}{2}$ by the Pythagorean Theorem.

Let $AP=x.$ It follows that $PM=\frac{\sqrt5}{2}-x.$ Applying the Pythagorean Theorem to right $\triangle ANP$ gives $NP^2=\left(\frac12\right)^2-x^2,$ and applying the Pythagorean Theorem to right $\triangle MNP$ gives $NP^2=1^2-\left(\frac{\sqrt5}{2}-x\right)^2.$ Equating the expressions for $NP^2$ produces \begin{align*} \left(\frac12\right)^2-x^2&=1^2-\left(\frac{\sqrt5}{2}-x\right)^2 \\ \frac14-x^2&=1-\frac54+\sqrt5x-x^2 \\ \frac12&=\sqrt5x. \end{align*} Finally, dividing both sides by $\sqrt5$ and then rationalizing the denominator, we obtain $x=\frac{1}{2\sqrt5}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.$

~MRENTHUSIASM

Solution 3 (Power of a Point)

Let circle $\omega$ intersect $\overline{AB}$ at point $N$. By Power of a Point, we have $AN^2=AP\cdot AM$. We know $AN=\frac{1}{2}$ because $N$ is the midpoint of $\overline{AB}$, and we can easily find $AM$ by the Pythagorean Theorem, which gives us $AM=\sqrt{1^2+\left(\frac{1}{2}\right)^2}=\frac{\sqrt{5}}{2}$. Our equation is now $\frac{1}{4}=AP\cdot \frac{\sqrt{5}}{2}$, or $AP=\frac{2}{4\sqrt{5}}=\frac{1}{2\sqrt{5}}=\frac{\sqrt{5}}{2\cdot 5}$, thus our answer is $\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.$

~Argonauts16

Solution 4 (Trigonometry)

Let $O$ be the center of the circle and the point of tangency between $\omega$ and $\overline{AD}$ be represented by $K$. We know that $\overline{AK} = \overline{KD} = \overline{DM} = \frac{1}{2}$. Consider the right triangle $\bigtriangleup ADM$. Let $\measuredangle AMD = \theta$.

Since $\omega$ is tangent to $\overline{DC}$ at $M$, $\measuredangle PMO = 90 - \theta$. Now, consider $\bigtriangleup POM$. This triangle is iscoceles because $\overline{PO}$ and $\overline{OM}$ are both radii of $\omega$. Therefore, $\measuredangle POM = 180 - 2(90 - \theta) = 2\theta$.

We can now use Law of Cosines on $\angle{POM}$ to find the length of ${PM}$ and subtract it from the length of ${AM}$ to find ${AP}$. Since $\cos{\theta} = \frac{1}{\sqrt{5}}$ and $\sin{\theta} = \frac{2}{\sqrt{5}}$, the double angle formula tells us that $\cos{2\theta} = -\frac{3}{5}$. We have \[PM^2 = \frac{1}{2} - \frac{1}{2}\cos{2\theta} \implies PM = \frac{2\sqrt{5}}{5}.\] By Pythagorean theorem, we find that $AM = \frac{\sqrt{5}}{2} \implies AP=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}$.

~awesome1st

Solution 5 (Trigonometry)

Take $O$ as the center and draw segment $ON$ perpendicular to $AM$, $ON\cap AM=N$, link $OM$. Then we have $OM\parallel AD$. So $\angle DAM=\angle OMA$. Since $AD=2AM=2OM=1$, we have $\cos\angle DAM=\cos\angle OMP=\frac{2}{\sqrt{5}}$. As a result, $NM=OM\cos\angle OMP=\frac{1}{2}\cdot \frac{2}{\sqrt{5}}=\frac{1}{\sqrt{5}}.$ Thus $PM=2NM=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}$. Since $AM=\frac{\sqrt{5}}{2}$, we have $AP=AM-PM=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}$.

~FANYUCHEN20020715

Solution 6 (Coordinate Geometry)

Place circle $\omega$ in the Cartesian plane such that the center lies on the origin. Then we can easily find the equation for $\omega$ as $x^2+y^2=\frac{1}{4}$, because it is not translated and the radius is $\frac{1}{2}$.

We have $A=\left(-\frac{1}{2}, \frac{1}{2}\right)$ and $M=\left(0, -\frac{1}{2}\right)$. The slope of the line passing through these two points is $\frac{\frac{1}{2}+\frac{1}{2}}{-\frac{1}{2}-0}=\frac{1}{-\frac{1}{2}}=-2$, and the $y$-intercept is simply $M$. This gives us the line passing through both points as $y=-2x-\frac{1}{2}$.

We substitute this into the equation for the circle to get $x^2+\left(-2x-\frac{1}{2}\right)^2=\frac{1}{4}$, or $x^2+4x^2+2x+\frac{1}{4}=\frac{1}{4}$. Simplifying gives $x(5x+2)=0$. The roots of this quadratic are $x=0$ and $x=-\frac{2}{5}$, but if $x=0$ we get point $M$, so we only want $x=-\frac{2}{5}$.

We plug this back into the linear equation to find $y=\frac{3}{10}$, and so $P=\left(-\frac{2}{5}, \frac{3}{10}\right)$. Finally, we use distance formula on $A$ and $P$ to get $AP=\sqrt{\left(-\frac{5}{10}+\frac{4}{10}\right)^2+\left(\frac{5}{10}-\frac{3}{10}\right)^2}=\sqrt{\frac{1}{100}+\frac{4}{100}}=\boxed{\textbf{(B) } \frac{\sqrt5}{10}}$.

~Argonauts16

Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/gd6VvCKMdu4

~Education, the Study of Everything




Video Solution by TheBeautyOfMath

https://youtu.be/6ujfjGLzVoE

See Also

2020 AMC 12B (Problems • Answer Key • Resources)
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