Difference between revisions of "2017 AMC 12B Problems/Problem 24"

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Problem 24

Quadrilateral $ABCD$ has right angles at $B$ and $C$ , $\triangle ABC \sim \triangle BCD$ , and $AB > BC$ . There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$ . What is $\tfrac{AB}{BC}$ ?

$\textbf{(A) } 1+\sqrt{2} \qquad \textbf{(B) } 2 + \sqrt{2} \qquad \textbf{(C) } \sqrt{17} \qquad \textbf{(D) } 2 + \sqrt{5} \qquad \textbf{(E) } 1 + 2\sqrt{3}$

Solution 1

Let $CD=1$ , $BC=x$ , and $AB=x^2$ . Note that $AB/BC=x$ . By the Pythagorean Theorem, $BD=\sqrt{x^2+1}$ . Since $\triangle BCD \sim \triangle ABC \sim \triangle CEB$ , the ratios of side lengths must be equal. Since $BC=x$ , $CE=\frac{x^2}{\sqrt{x^2+1}}$ and $BE=\frac{x}{\sqrt{x^2+1}}$ . Let F be a point on $\overline{BC}$ such that $\overline{EF}$ is an altitude of triangle $CEB$ . Note that $\triangle CEB \sim \triangle CFE \sim \triangle EFB$ . Therefore, $BF=\frac{x}{x^2+1}$ and $CF=\frac{x^3}{x^2+1}$ . Since $\overline{CF}$ and $\overline{BF}$ form altitudes of triangles $CED$ and $BEA$ , respectively, the areas of these triangles can be calculated. Additionally, the area of triangle $BEC$ can be calculated, as it is a right triangle. Solving for each of these yields: $[BEC]=[CED]=[BEA]=(x^3)/(2(x^2+1))$ $[ABCD]=[AED]+[DEC]+[CEB]+[BEA]$ $(AB+CD)(BC)/2= 17*[CEB]+ [CEB] + [CEB] + [CEB]$ $(x^3+x)/2=(20x^3)/(2(x^2+1))$ $(x)(x^2+1)=20x^3/(x^2+1)$ $(x^2+1)^2=20x^2$ $x^4-18x^2+1=0 \implies x^2=9+4\sqrt{5}=4+2(2\sqrt{5})+5$ Therefore, the answer is $\boxed{\textbf{(D) } 2+\sqrt{5}}$

Solution 2

Draw line $FG$ through $E$ , with $F$ on $BC$ and $G$ on $AD$ , $FG \parallel AB$ . WLOG let $CD=1$ , $CB=x$ , $AB=x^2$ . By weighted average $FG=\frac{1+x^4}{1+x^2}$ .

Meanwhile, $FE:EG=[\triangle CBE]:[\triangle ADE]=1:17$ .

$FE=\frac{x^2}{1+x^2}$ . We obtain $\frac{1+x^4}{1+x^2}=\frac{18x^2}{1+x^2}$ , namely $x^4-18x^2+1=0$ .

The rest is the same as Solution 1.

Solution 3

Let $A=(-1,4a), B=(-1,0), C=(1,0), D=\bigg(1,\frac{1}{a}\bigg)$ . Then from the similar triangles condition, we compute $CE=\frac{4a}{\sqrt{4a^2+1}}$ and $BE=\frac{2}{\sqrt{4a^2+1}}$ . Hence, the $y$ -coordinate of $E$ is just $\frac{BE\cdot CE}{BC}=\frac{4a}{4a^2+1}$ . Since $E$ lies on the unit circle, we can compute the $x$ coordinate as $\frac{1-4a^2}{4a^2+1}$ . By Shoelace, we want $\frac{1}{2}\det\begin{bmatrix} -1 & 4a & 1\\ \frac{1-4a^2}{4a^2+1} & \frac{4a}{4a^2+1} & 1\\ 1 & \frac{1}{a} & 1 \end{bmatrix}=17\cdot\frac{1}{2}\cdot 2 \cdot \frac{4a}{4a^2+1}$ Factoring out denominators and expanding by minors, this is equivalent to $\frac{32a^4-8a^2+2}{2a(4a^2+1)}=\frac{68a}{4a^2+1} \Longrightarrow 16a^4-72a^2+1=0$ This factors as $(4a^2-8a-1)(4a^2+8a-1)=0$ , so $a=1+\frac{\sqrt{5}}{2}$ and so the answer is $\textbf{(D) \ }$ .

@@ Line 27: / Line 27: @@
 The rest is the same as Solution 1.
+== Solution 3==
+Let <math>A=(-1,4a), B=(-1,0), C=(1,0), D=\bigg(1,\frac{1}{a}\bigg)</math>. Then from the similar triangles condition, we compute <math>CE=\frac{4a}{\sqrt{4a^2+1}}</math> and <math>BE=\frac{2}{\sqrt{4a^2+1}}</math>. Hence, the <math>y</math>-coordinate of <math>E</math> is just <math>\frac{BE\cdot CE}{BC}=\frac{4a}{4a^2+1}</math>. Since <math>E</math> lies on the unit circle, we can compute the <math>x</math> coordinate as <math>\frac{1-4a^2}{4a^2+1}</math>. By Shoelace, we want <cmath>\frac{1}{2}\det\begin{bmatrix}
+-1 & 4a & 1\\
+\frac{1-4a^2}{4a^2+1} & \frac{4a}{4a^2+1} & 1\\
+& \frac{1}{a} & 1
+\end{bmatrix}=17\cdot\frac{1}{2}\cdot 2 \cdot \frac{4a}{4a^2+1}</cmath>Factoring out denominators and expanding by minors, this is equivalent to
+<cmath>\frac{32a^4-8a^2+2}{2a(4a^2+1)}=\frac{68a}{4a^2+1} \Longrightarrow 16a^4-72a^2+1=0</cmath>This factors as <math>(4a^2-8a-1)(4a^2+8a-1)=0</math>, so <math>a=1+\frac{\sqrt{5}}{2}</math> and so the answer is <math> \textbf{(D) \ }</math>.
 ==See Also==
 {{AMC12 box|year=2017|ab=B|num-b=23|num-a=25}}
 {{MAA Notice}}

2017 AMC 12B (Problems • Answer Key • Resources)
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