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21

Part of 2015 AMC 8

Problems(1)

Equiangular Hexagons

Source: 2015 AMC 8 #21

11/25/2015
In the given figure hexagon ABCDEFABCDEF is equiangular, ABJIABJI and FEHGFEHG are squares with areas 1818 and 3232 respectively, △JBK\triangle JBK is equilateral and FE=BCFE=BC. What is the area of △KBC\triangle KBC?
<spanclass=′latex−bold′>(A)</span>62<spanclass=′latex−bold′>(B)</span>9<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>92<spanclass=′latex−bold′>(E)</span>32<span class='latex-bold'>(A) </span>6\sqrt{2}\qquad<span class='latex-bold'>(B) </span>9\qquad<span class='latex-bold'>(C) </span>12\qquad<span class='latex-bold'>(D) </span>9\sqrt{2}\qquad<span class='latex-bold'>(E) </span>32
[asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label("II",(-4,8+6*sqrt(2)),dir(100)); label("JJ",(4,8+6*sqrt(2)),dir(80)); label("AA",(-4,6*sqrt(2)),dir(280)); label("BB",(4,6*sqrt(2)),dir(250)); label("CC",(8,0),W); label("DD",(4,-6*sqrt(2)),NW); label("EE",(-4,-6*sqrt(2)),NE); label("FF",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label("KK",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label("HH",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label("GG",(-8-6*sqrt(2),-4),W); label("3232",(-10,-8),N); label("1818",(0,6*sqrt(2)+2),N); [/asy]
AMC 8geometry