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

Revision as of 09:11, 7 October 2023

Problem

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, $FE:EG=[\triangle CBE]:[\triangle ADE]=1:17$ . This follows from comparing the ratios of triangle DEG to CFE and triangle AEG to FEB, both pairs in which the two triangles share a height perpendicular to FG, and have base ratio $EG:FE$ .

$FE=\frac{x^2}{1+x^2}$ . 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

Diagram by isabelchen

Let $AB = a$ , $BC = b$ , $AC = \sqrt{a^2+b^2}$

Note that $E$ cannot be the intersection of $AC$ and $BD$ , as that would mean $[AED] = [CEB]$

$5+9=\boxed{\textbf{(B) } 14}$ .

~isabelchen

Solution 4

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 $\textbf{(D) \ }$ .

Solution 5

Let $C = (0,0), D=(\frac1a, 0), B = (0,1), A = (a,1)$ where $a>1$ . Because $BC = 1, a = \frac{AB}{BC}$ . Notice that the diagonals are perpendicular with slopes of $\frac1a$ and $-a$ . Let the intersection of $AC$ and $BD$ be $F$ , then $\triangle BFC \sim \triangle ABC$ . However, because $ABCD$ is a trapezoid, $\triangle$ $BCF$ and $\triangle ADF$ share the same area, therefore $\triangle$ $BCE$ is the reflection of $\triangle$ $BCF$ over the perpendicular bisector of $BC$ , which is $y=\frac12$ . We use the linear equations of the diagonals, $y = -ax + 1, y = \frac1a x$ , to find the coordinates of $F$ . $-ax+1 = \frac1ax \Longrightarrow x = \frac{1}{a+\frac1a} = \frac{a}{a^2+1}$ $y = \frac1ax = \frac{1}{a^2+1}$ The y-coordinate of $E$ is simply $1-\frac{1}{a^2+1} = \frac{a^2}{a^2+1}$ The area of $\triangle BCE$ is $\frac12 \frac{a}{a^2+1}$ . We apply shoelace theorem to solve for the area of $\triangle ADE$ . The coordinates of the triangle are $\{(\frac{a}{a^2+1}, \frac{a^2}{a^2+1}), (a,1), (\frac1a, 0)\}$ , so the area is

$\frac12 |\frac{a^3}{a^2+1} + \frac1a - \frac{a}{a^2+1} - \frac{a}{a^2+1}| = \frac12 |\frac{a^3-2a}{a^2+1} + \frac1a|$ $= \frac12 |\frac{a^4-2a^2}{a(a^2+1)} + \frac{a^2+1}{a(a^2+1)}| = \frac12 \frac{a^4-a^2+1}{a(a^2+1)}$ Finally, we use the property that the ratio of areas equals $17$ $\frac{\frac12}{\frac12} \frac{\frac{a^4-a^2+1}{a(a^2+1)}}{\frac{a}{a^2+1}} = 17 \Rightarrow \frac{a^4-a^2+1}{a^2} = 17 \Rightarrow a^4-18a^2+1 = 0$ $a^2 = 9+4\sqrt{5} = (2+\sqrt{5})^2 \Longrightarrow a = \boxed{\textbf{(D) } 2+\sqrt{5}}$

~Zeric

Video Solution by MOP 2024

https://youtu.be/h92s2BxlohI

~r00tsOfUnity

Notes

1) $\sqrt{17}$ is the most relevant answer choice because it shares numbers with the givens of the problem.

2) It's a very good guess to replace finding the area of triangle AED with the area of the triangle DAF, where F is the projection of D onto AB(then find the closest answer choice).

@@ Line 28: / Line 28: @@
 The rest is the same as Solution 1.
-== Solution 3==
+==Solution 3==
+Diagram by [https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+Let <math>AB = a</math>, <math>BC = b</math>, <math>AC = \sqrt{a^2+b^2}</math>
+Note that <math>E</math> cannot be the intersection of <math>AC</math> and <math>BD</math>, as that would mean <math>[AED] = [CEB]</math>
+<math>5+9=\boxed{\textbf{(B) } 14}</math>.
+~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
+== Solution 4==
 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\\
@@ Line 36: / Line 49: @@
 <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>.
-==Solution 4==
+==Solution 5==
 Let <math>C = (0,0), D=(\frac1a, 0), B = (0,1), A = (a,1)</math> where <math>a>1</math>. Because <math>BC = 1, a = \frac{AB}{BC}</math>. Notice that the diagonals are perpendicular with slopes of <math>\frac1a</math> and <math>-a</math>. Let the intersection of <math>AC</math> and <math>BD</math> be <math>F</math>, then <math>\triangle BFC \sim \triangle ABC</math>. However, because <math>ABCD</math> is a trapezoid, <math>\triangle</math><math>BCF</math> and <math>\triangle ADF</math> share the same area, therefore <math>\triangle</math><math>BCE</math> is the reflection of <math>\triangle</math><math>BCF</math> over the perpendicular bisector of <math>BC</math>, which is <math>y=\frac12</math>. We use the linear equations of the diagonals, <math>y = -ax + 1, y = \frac1a x</math>, to find the coordinates of <math>F</math>. <cmath>-ax+1 = \frac1ax \Longrightarrow x = \frac{1}{a+\frac1a} = \frac{a}{a^2+1}</cmath> <cmath>y = \frac1ax = \frac{1}{a^2+1}</cmath>

2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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