|
|
| (23 intermediate revisions by 9 users not shown) |
| Line 1: |
Line 1: |
| − | ==Problem==
| + | #redirect [[2022 AMC 10A Problems/Problem 23]] |
| − | Isosceles trapezoid <math>ABCD</math> has parallel sides <math>\overline{AD}</math> and <math>\overline{BC},</math> with <math>BC < AD</math> and <math>AB = CD.</math> There is a point <math>P</math> in the plane such that <math>PA=1, PB=2, PC=3,</math> and <math>PD=4.</math> What is <math>\tfrac{BC}{AD}?</math>
| |
| − | | |
| − | <math>\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}</math>
| |
| − | | |
| − | ==Solution==
| |
| − | Consider the reflection <math>P^{\prime}</math> of <math>P</math> over the perpendicular bisector of <math>\overline{BC}</math>, creating two new isosceles trapezoids <math>DAPP^{\prime}</math> and <math>CBPP^{\prime}</math>. Under this reflection, <math>P^{\prime}A=PD=4</math>, <math>P^{\prime}D=PA=1</math>, <math>P^{\prime}C=PB=2</math>, and <math>P^{\prime}B=PC=3</math>. By Ptolmey's theorem <cmath>\begin{align*} PP^{\prime}\cdot AD+1=16 \\ PP^{\prime}\cdot BC+4=9\end{align*}</cmath> Thus <math>PP^{\prime}\cdot AD=15</math> and <math>PP^{\prime}\cdot BC=5</math>; dividing these two equations and taking the reciprocal yields <math>\frac{BC}{AD}=\boxed{\textbf{(B)}~\frac{1}{3}}</math>.
| |
| − | | |
| − | ==See also==
| |
| − | {{AMC12 box|year=2022|ab=A|num-b=19|num-a=21}}
| |
| − | {{AMC10 box|year=2022|ab=A|num-b=22|num-a=24}}
| |
| − | | |
| − | [[Category:Intermediate Geometry Problems]] | |
| − | {{MAA Notice}}
| |
