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Difference between revisions of "2020 AMC 12B Problems/Problem 10"

m (Written solutions up front.)
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==Problem==
 
==Problem==
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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>
 +
 +
==Diagram==
 +
[[File:2020 AMC 12B Problem 12 Diagram.png]]
 +
 +
~MRENTHUSIASM (by Geometry Expressions)
  
 
==Solution 1 (Angle Chasing/Trig)==
 
==Solution 1 (Angle Chasing/Trig)==
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~awesome1st
 
~awesome1st
−
==Solution 2(Coordinate Bash)==
+
 
 +
==Solution 2 (Coordinate Bash)==
  
 
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>.  
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~Argonauts16
 
~Argonauts16
  
−
==Solution 3(Power of a Point)==
+
==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  
 
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>
 
<math>\boxed{\textbf{(B) } \frac{\sqrt5}{10}}.</math>
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~QIDb602
 
~QIDb602
  
−
==Solution 6 intersecting chords==
+
==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>
 
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>MQ=AQ=\frac{1}{2}AM = \frac{\sqrt{5}}{4}</math> and <math>NQ=\frac{1}{4}</math>

Revision as of 07:59, 22 April 2021

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

File:2020 AMC 12B Problem 12 Diagram.png

~MRENTHUSIASM (by Geometry Expressions)

Solution 1 (Angle Chasing/Trig)

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 $AM = \frac{\sqrt{5}}{2} \implies \boxed{\textbf{(B) } \frac{\sqrt5}{10}}$

~awesome1st

Solution 2 (Coordinate Bash)

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 $AP=\sqrt{\left(-\frac{5}{10}+\frac{4}{10}\right)^2+\left(\frac{5}{10}-\frac{3}{10}\right)^2}=\sqrt{\frac{1}{100}+\frac{4}{100}}=\boxed{\mathbf{(B) } \frac{\sqrt{5}}{10}}$.

~Argonauts16

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

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 $AP=AM-PM=\frac{\sqrt{5}}{10}$. Put $\boxed{B}$.

~FANYUCHEN20020715

Solution 5 (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] 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, $AM=\frac{\sqrt{5}}{2}$. Now we just need to find $AP$ using similar triangles. \[\triangle APN\sim\triangle ANM\Rightarrow\frac{AP}{AN}=\frac{AN}{AM}\Rightarrow\frac{AP}{\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{\sqrt{5}}{2}}\Rightarrow AP=\boxed{\textbf{(B) }\frac{\sqrt{5}}{10}}\] ~QIDb602

Solution 6 (Intersecting Chords)

Label the midpoint of $AD$ as $N$, and let $Q$ be the intersection of $ON$ and $AM$ Then $MQ=AQ=\frac{1}{2}AM = \frac{\sqrt{5}}{4}$ and $NQ=\frac{1}{4}$

Then $\frac{1}{4}*\frac{3}{4}=PQ*QM$ and $AP=AQ-PQ$

~ERMSCoach

Video Solution

https://youtu.be/6ujfjGLzVoE

~IceMatrix

See Also

2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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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