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Difference between revisions of "2022 AMC 12A Problems/Problem 20"
(→‎Solution 2 (Coordinate Bashing))
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==Solution 2 (Coordinate Bashing)==
 
==Solution 2 (Coordinate Bashing)==
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Since we're given distances and nothing else, we can represent each point as a coordinate and use the distance formula to set up a series of systems and equations.
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Let the height of the trapezoid be <math>h</math>, and let the coordinates of <math>A</math> and <math>D</math> be at <math>(-a,0)</math> and <math>(a,0)</math>, respectively. Then let <math>B</math> and <math>C</math> be at <math>(-b,h)</math> and <math>(b,h)</math>, respectively. This follows the rules that this is an isosceles trapezoid since the origin is centered on the middle of <math>AD</math>. Finally, let <math>P</math> be located at point <math>(c,d)</math>.
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The distance from <math>P</math> to <math>A</math> is <math>1</math>, so by the distance formula: <cmath>\sqrt{(c+a)^2+(d-h)^2} = 1 \implies (c+a)^2+(d-h)^2 = 1</cmath>
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The distance from <math>P</math> to <math>D</math> is <math>4</math>, so <cmath>\sqrt{(c-a)^2+(d-h)^2} = 1 \implies (c-a)^2+(d-h)^2 = 16</cmath>
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Looking at these two equations alone, notice that the second term is the same for both equations, so we can subtract the equations. This yields <cmath>-4ac = 15</cmath>
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Next, the distance from <math>P</math> to <math>B</math> is <math>2</math>, so <cmath>\sqrt{(c+b)^2+(d-h)^2} = 2 \implies (c+b)^2+(d-h)^2 = 4</cmath>
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The distance from <math>P</math> to <math>C</math> is <math>3</math>, so <cmath>\sqrt{(c-b)^2+(d-h)^2} = 3 \implies (c-b)^2+(d-h)^2 = 9</cmath>
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Again, we can subtract these equations, yielding <cmath>-4bc = 5</cmath>
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We can now divide the equations to eliminate <math>c</math>, yielding <cmath>\frac{b}{a} = \frac{5}{15} = \frac{1}{3}</cmath>
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We wanted to find <math>\frac{BC}{AD}</math>. But since <math>b</math> is half of <math>BC</math> and <math>a</math> is half of <math>AD</math>, this ratio is equal to the ratio we want.
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Therefore <math>\frac{BC}{AD} = \frac{1}{3} = \boxed{B}</math>
  
 
Solution in Progress
 
Solution in Progress

Revision as of 14:04, 12 November 2022

Problem

Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$

$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$

Solution 1

Consider the reflection $P^{\prime}$ of $P$ over the perpendicular bisector of $\overline{BC}$, creating two new isosceles trapezoids $DAPP^{\prime}$ and $CBPP^{\prime}$. Under this reflection, $P^{\prime}A=PD=4$, $P^{\prime}D=PA=1$, $P^{\prime}C=PB=2$, and $P^{\prime}B=PC=3$. By Ptolmey's theorem \begin{align*} PP^{\prime}\cdot AD+1=16 \\ PP^{\prime}\cdot BC+4=9\end{align*} Thus $PP^{\prime}\cdot AD=15$ and $PP^{\prime}\cdot BC=5$; dividing these two equations and taking the reciprocal yields $\frac{BC}{AD}=\boxed{\textbf{(B)}~\frac{1}{3}}$.

Solution 2 (Coordinate Bashing)

Since we're given distances and nothing else, we can represent each point as a coordinate and use the distance formula to set up a series of systems and equations. Let the height of the trapezoid be $h$, and let the coordinates of $A$ and $D$ be at $(-a,0)$ and $(a,0)$, respectively. Then let $B$ and $C$ be at $(-b,h)$ and $(b,h)$, respectively. This follows the rules that this is an isosceles trapezoid since the origin is centered on the middle of $AD$. Finally, let $P$ be located at point $(c,d)$.

The distance from $P$ to $A$ is $1$, so by the distance formula: \[\sqrt{(c+a)^2+(d-h)^2} = 1 \implies (c+a)^2+(d-h)^2 = 1\] The distance from $P$ to $D$ is $4$, so \[\sqrt{(c-a)^2+(d-h)^2} = 1 \implies (c-a)^2+(d-h)^2 = 16\]

Looking at these two equations alone, notice that the second term is the same for both equations, so we can subtract the equations. This yields \[-4ac = 15\]

Next, the distance from $P$ to $B$ is $2$, so \[\sqrt{(c+b)^2+(d-h)^2} = 2 \implies (c+b)^2+(d-h)^2 = 4\] The distance from $P$ to $C$ is $3$, so \[\sqrt{(c-b)^2+(d-h)^2} = 3 \implies (c-b)^2+(d-h)^2 = 9\]

Again, we can subtract these equations, yielding \[-4bc = 5\]

We can now divide the equations to eliminate $c$, yielding \[\frac{b}{a} = \frac{5}{15} = \frac{1}{3}\]

We wanted to find $\frac{BC}{AD}$. But since $b$ is half of $BC$ and $a$ is half of $AD$, this ratio is equal to the ratio we want.

Therefore $\frac{BC}{AD} = \frac{1}{3} = \boxed{B}$

Solution in Progress

~KingRavi

Solution 3 (Cheese)

Notice that the question never says what the height of the trapezoid is; the only property we know about it is that $AC=BD$. Therefore, we can say WLOG that the height of the trapezoid is $0$ and all $5$ points, including $P$, lie on the same line with $PA=AB=BC=CD=1$. Notice that this satisfies the problem requirements because $PA=1, PB=2, PC=3,PD=4$, and $AC=BD=2$. Now all we have to find is $\frac{BC}{AD} = \frac{1}{3}= \boxed{B}$

~KingRavi

Video Solution By ThePuzzlr

https://youtu.be/KqtpaHy-eoU

~ MathIsChess


Video Solution

https://youtu.be/hvIOvjjQvIw

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution by OmegaLearn using Pythagorean Theorem

https://youtu.be/jnm2alniaM4

~ pi_is_3.14

See also

2022 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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
2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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 10 Problems and Solutions

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