MathDB
Find [ABF]

Source: 1978 AHSME Problem 23

June 18, 2014
geometryanalytic geometryratiotrigonometryAMC

Problem Statement

[asy] size(100); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,1)--(1,0)); draw((0,0)--(.5,sqrt(3)/2)--(1,0)); label("AA",(0,0),SW); label("BB",(1,0),SE); label("CC",(1,1),NE); label("DD",(0,1),NW); label("EE",(.5,sqrt(3)/2),E); label("FF",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W); //Credit to chezbgone2 for the diagram[/asy]
Vertex EE of equilateral triangle ABEABE is in the interior of square ABCDABCD, and FF is the point of intersection of diagonal BDBD and line segment AEAE. If length ABAB is 1+3\sqrt{1+\sqrt{3}} then the area of ABF\triangle ABF is
<spanclass=latexbold>(A)</span>1<spanclass=latexbold>(B)</span>22<spanclass=latexbold>(C)</span>32<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>\frac{\sqrt{2}}{2}\qquad<span class='latex-bold'>(C) </span>\frac{\sqrt{3}}{2}
<spanclass=latexbold>(D)</span>423<spanclass=latexbold>(E)</span>12+34\qquad<span class='latex-bold'>(D) </span>4-2\sqrt{3}\qquad <span class='latex-bold'>(E) </span>\frac{1}{2}+\frac{\sqrt{3}}{4}