Difference between revisions of "2018 AMC 12A Problems/Problem 20"

Revision as of 01:32, 24 December 2018

Problem

Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$ . Let $M$ be the midpoint of hypotenuse $\overline{BC}$ . Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$ , respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$ , the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$ , where $a$ , $b$ , and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$ ?

$\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }11 \qquad \textbf{(D) }12 \qquad \textbf{(E) }13 \qquad$

Solution 1

Observe that $\triangle{EMI}$ is isosceles right ( $M$ is the midpoint of diameter arc $EI$ ), so $MI=2,MC=\frac{3}{\sqrt{2}}$ . With $\angle{MCI}=45^\circ$ , we can use Law of Cosines to determine that $CI=\frac{3\pm\sqrt{7}}{2}$ . The same calculations hold for $BE$ also, and since $CI<BE$ , we deduce that $CI$ is the smaller root, giving the answer of $\boxed{12}$ . (trumpeter)

Solution 2 (Using Ptolemy)

We first claim that $\triangle{EMI}$ is isosceles and right.

Proof: Construct $\overline{MF}\perp\overline{AB}$ and $\overline{MG}\perp\overline{AC}$ . Since $\overline{AM}$ bisects $\angle{BAC}$ , one can deduce that $MF=MG$ . Then by AAS it is clear that $MI=ME$ and therefore $\triangle{EMI}$ is isosceles. Since quadrilateral $AIME$ is cyclic, one can deduce that $\angle{EMI}=90^\circ$ . Q.E.D.

Since the area of $\triangle{EMI}$ is 2, we can find that $MI=ME=2$ , $EI=2\sqrt{2}$

Since $M$ is the mid-point of $\overline{BC}$ , it is clear that $AM=\frac{3\sqrt{2}}{2}$ .

Now let $AE=a$ and $AI=b$ . By Ptolemy's Theorem, in cyclic quadrilateral $AIME$ , we have $2a+2b=6$ . By Pythagorean Theorem, we have $a^2+b^2=8$ . One can solve the simultaneous system and find $b=\frac{3+\sqrt{7}}{2}$ . Then by deducting the length of $\overline{AI}$ from 3 we get $CI=\frac{3-\sqrt{7}}{2}$ , giving the answer of $\boxed{12}$ . (Surefire2019)

Solution 3 (More Elementary)

Like above, notice that $\triangle{EMI}$ is isosceles and right, which means that $\dfrac{ME \cdot MI}{2} = 2$ , so $MI^2=4$ and $MI = 2$ . Then construct $\overline{MF}\perp\overline{AB}$ and $\overline{MG}\perp\overline{AC}$ as well as $\overline{MI}$ . It's clear that $MG^2+GI^2 = MI^2$ by Pythagorean, so knowing that $MG = \dfrac{AB}{2} = \dfrac{3}{2}$ allows one to solve to get $GI = \dfrac{\sqrt{7}}{2}$ . By just looking at the diagram, $CI=AC-MF-GI=\dfrac{3-\sqrt{7}}{2}$ . The answer is thus $3+7+2=12$ .

Solution 4 (Coordinate Geometry)

Let $A$ lie on $(0,0)$ , $E$ on $(0,y)$ , $I$ on $(x,0)$ , and $M$ on $(\frac{3}{2},\frac{3}{2})$ . Since ${AIME}$ is cyclic, $\angle EMI$ (which is opposite of another right angle) must be a right angle; therefore, $\vec{ME} \cdot \vec{MI} = <\frac{-3}{2}, y-\frac{3}{2}> \cdot <x-\frac{3}{2}, -\frac{3}{2}> = 0$ . Compute the dot product to arrive at the relation $y=3-x$ . We can set up another equation involving the area of $\triangle EMI$ using the Shoelace Theorem. This is $2=(\frac{1}{2})[(\frac{3}{2})(y-\frac{3}{2})+(x)(-y)+(x+\frac{3}{2})(\frac{3}{2})]$ . Multiplying, substituting $3-x$ for $y$ , and simplifying, we get $x^2 -3x + \frac{1}{2}=0$ . Thus, $(x,y)=$ $(\frac{3 \pm \sqrt{7}}{2},\frac{3 \mp \sqrt{7}}{2})$ . But $AI>AE$ , meaning $x=AI=\frac{3 + \sqrt{7}}{2} \rightarrow CI = 3-\frac{3 + \sqrt{7}}{2}=\frac{3 - \sqrt{7}}{2}$ , and the final answer is $3+7+2=\boxed{12}$ .

Solution 5 (Quick)

From $AIME$ cyclic we get $\angle{MEI} = \angle{MAI} = 45^\circ$ and $\angle{MIE} = \angle{MAE} = 45^\circ$ , so $\triangle{EMI}$ is an isosceles right triangle. From $[EMI]=2$ we get $EM=MI=2$ . Notice $\triangle{AEM} \cong \triangle{CIM}$ , because $\angle{AEM}=180-\angle{AIM}=\angle{CIM}$ , $EM=IM$ , and $\angle{EAM}=\angle{ICM}=45^\circ$ . Let $CI=AE=x$ , so $AI=3-x$ . By Pythagoras on $\triangle{EAI}$ we have $x^2+(3-x)^2=EI^2=8$ , and solve this to get $x=CI=\dfrac{3-\sqrt{7}}{2}$ for a final answer of $3+7+2=\boxed{12}$ .

@@ Line 36: / Line 36: @@
 == Solution 5 (Quick) ==
-From <math>AIME</math> cyclic we get <math>\angle{MEI} = \angle{MAI} = 45^\circ</math> and <math>\angle{MIE} = \angle{MAE} = 45^\circ</math>, so <math>\triangle{EMI}</math> is an isosceles right triangle and from <math>[EMI]=2</math> we get <math>EM=MI=2</math>. Notice <math>\triangle{AEM} \cong \triangle{CIM}</math>, because <math>\angle{AEM}=180-\angle{AIM}=\angle{CIM}</math>, <math>EM=IM</math>, and <math>\angle{EAM}=\angle{ICM}=45^\circ</math>. Let <math>CI=AE=x</math>, so <math>AI=3-x</math>. By Pythagoras on <math>\triangle{EAI}</math> we have <math>x^2+(3-x)^2=EI^2=8</math>, and solve this to get <math>x=CI=\dfrac{3-\sqrt{7}}{2}</math> for a final answer of <math>3+7+2=\boxed{12}</math>.
+From <math>AIME</math> cyclic we get <math>\angle{MEI} = \angle{MAI} = 45^\circ</math> and <math>\angle{MIE} = \angle{MAE} = 45^\circ</math>, so <math>\triangle{EMI}</math> is an isosceles right triangle.
+From <math>[EMI]=2</math> we get <math>EM=MI=2</math>.
+Notice <math>\triangle{AEM} \cong \triangle{CIM}</math>, because <math>\angle{AEM}=180-\angle{AIM}=\angle{CIM}</math>, <math>EM=IM</math>, and <math>\angle{EAM}=\angle{ICM}=45^\circ</math>.
+Let <math>CI=AE=x</math>, so <math>AI=3-x</math>.
+By Pythagoras on <math>\triangle{EAI}</math> we have <math>x^2+(3-x)^2=EI^2=8</math>, and solve this to get <math>x=CI=\dfrac{3-\sqrt{7}}{2}</math> for a final answer of <math>3+7+2=\boxed{12}</math>.
 ==See Also==
 {{AMC12 box|year=2018|ab=A|num-b=19|num-a=21}}
 {{MAA Notice}}

2018 AMC 12A (Problems • Answer Key • Resources)
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