MathDB

Let

ABCD

be an isosceles trapezoid with

\overline{BC}\parallel \overline{AD}

and

AB=CD

. Points

X

and

Y

lie on diagonal

\overline{AC}

with

X

between

A

and

Y

, as shown in the figure. Suppose

\angle AXD = \angle BYC = 90^\circ

AX = 3

XY = 1

, and

YC = 2

. What is the area of

ABCD?

[asy] size(6cm); usepackage("mathptmx"); import geometry; void perp(picture pic=currentpicture, pair O, pair M, pair B, real size=5, pen p=currentpen, filltype filltype = NoFill){ perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype); } pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6)); draw(A--B--C--D--cycle,p); draw(A--C,p); draw(B--Y,p); draw(D--X,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(X,q); dot(Y,q); label("2",C--Y,S); label("1",Y--X,S); label("3",X--A,S); label("

A

",A,E); label("

B

",B,N); label("

C

",C,W); label("

D

",D,S); label("

Y

",Y,sqrt(2)*NE); label("

X

",X,N); perp(B,Y,C,8,p); perp(A,X,D,8,p); [/asy]

<span class='latex-bold'>(A)</span>\: 15\qquad<span class='latex-bold'>(B)</span> \: 5\sqrt{11}\qquad<span class='latex-bold'>(C)</span> \: 3\sqrt{35}\qquad<span class='latex-bold'>(D)</span> \: 18\qquad<span class='latex-bold'>(E)</span> \: 7\sqrt{7}

Problem Statement