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MATH / In the diagram, AB, BC, CD, DE, EF, FG, GH, and HK all have length 4, and all angles are right angles, with the exception of the angles at D and F. [asy] draw((0,0)--(0,4)--(4,4)--…

Problem

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problem

In the diagram, AB,AB, BC,BC, CD,CD, DE,DE, EF,EF, FG,FG, GH,GH, and HKHK all have length 4,4, and all angles are right angles, with the exception of the angles at DD and F.F. [asy] draw((0,0)--(0,4)--(4,4)--(4,8)--(6.8284,5.1716)--(9.6569,8)--(9.6569,4)--(13.6569,4)--(13.6569,0)--cycle,black+linewidth(1)); draw((0,0)--(0.5,0)--(0.5,0.5)--(0,0.5)--cycle,black+linewidth(1)); draw((0,4)--(0.5,4)--(0.5,3.5)--(0,3.5)--cycle,black+linewidth(1)); draw((4,4)--(4,4.5)--(3.5,4.5)--(3.5,4)--cycle,black+linewidth(1)); draw((6.8284,5.1716)--(7.0784,5.4216)--(6.8284,5.6716)--(6.5784,5.4216)--cycle,black+linewidth(1)); draw((9.6569,4)--(10.1569,4)--(10.1569,4.5)--(9.6569,4.5)--cycle,black+linewidth(1)); draw((13.6569,4)--(13.1569,4)--(13.1569,3.5)--(13.6569,3.5)--cycle,black+linewidth(1)); draw((13.6569,0)--(13.1569,0)--(13.1569,0.5)--(13.6569,0.5)--cycle,black+linewidth(1)); label("AA",(0,0),W); label("BB",(0,4),NW); label("CC",(4,4),S); label("DD",(4,8),N); label("EE",(6.8284,5.1716),S); label("FF",(9.6569,8),N); label("GG",(9.6569,4),S); label("HH",(13.6569,4),NE); label("KK",(13.6569,0),E); [/asy] Determine the length of DF.DF. [asy] draw((0,0)--(2.8284,-2.8284)--(5.6568,0),black+linewidth(1)); draw((0,0)--(5.6568,0),black+linewidth(1)+dashed); draw((2.8284,-2.8284)--(3.0784,-2.5784)--(2.8284,-2.3284)--(2.5784,-2.5784)--cycle,black+linewidth(1)); label("DD",(0,0),N); label("EE",(2.8284,-2.8284),S); label("FF",(5.6568,0),N); [/asy]
Plain-text mathematical notation (without MathML)
In the diagram, AB, BC, CD, DE, EF, FG, GH, and HK all have length 4, and all angles are right angles, with the exception of the angles at D and F.

[asy]
draw((0,0)--(0,4)--(4,4)--(4,8)--(6.8284,5.1716)--(9.6569,8)--(9.6569,4)--(13.6569,4)--(13.6569,0)--cycle,black+linewidth(1));
draw((0,0)--(0.5,0)--(0.5,0.5)--(0,0.5)--cycle,black+linewidth(1));
draw((0,4)--(0.5,4)--(0.5,3.5)--(0,3.5)--cycle,black+linewidth(1));
draw((4,4)--(4,4.5)--(3.5,4.5)--(3.5,4)--cycle,black+linewidth(1));
draw((6.8284,5.1716)--(7.0784,5.4216)--(6.8284,5.6716)--(6.5784,5.4216)--cycle,black+linewidth(1));
draw((9.6569,4)--(10.1569,4)--(10.1569,4.5)--(9.6569,4.5)--cycle,black+linewidth(1));
draw((13.6569,4)--(13.1569,4)--(13.1569,3.5)--(13.6569,3.5)--cycle,black+linewidth(1));
draw((13.6569,0)--(13.1569,0)--(13.1569,0.5)--(13.6569,0.5)--cycle,black+linewidth(1));
label("A",(0,0),W);
label("B",(0,4),NW);
label("C",(4,4),S);
label("D",(4,8),N);
label("E",(6.8284,5.1716),S);
label("F",(9.6569,8),N);
label("G",(9.6569,4),S);
label("H",(13.6569,4),NE);
label("K",(13.6569,0),E);
[/asy]

Determine the length of DF.

[asy]
draw((0,0)--(2.8284,-2.8284)--(5.6568,0),black+linewidth(1));
draw((0,0)--(5.6568,0),black+linewidth(1)+dashed);
draw((2.8284,-2.8284)--(3.0784,-2.5784)--(2.8284,-2.3284)--(2.5784,-2.5784)--cycle,black+linewidth(1));
label("D",(0,0),N);
label("E",(2.8284,-2.8284),S);
label("F",(5.6568,0),N);
[/asy]
Original LaTeX notation
In the diagram, $AB,$ $BC,$ $CD,$ $DE,$ $EF,$ $FG,$ $GH,$ and $HK$ all have length $4,$ and all angles are right angles, with the exception of the angles at $D$ and $F.$

[asy]
draw((0,0)--(0,4)--(4,4)--(4,8)--(6.8284,5.1716)--(9.6569,8)--(9.6569,4)--(13.6569,4)--(13.6569,0)--cycle,black+linewidth(1));
draw((0,0)--(0.5,0)--(0.5,0.5)--(0,0.5)--cycle,black+linewidth(1));
draw((0,4)--(0.5,4)--(0.5,3.5)--(0,3.5)--cycle,black+linewidth(1));
draw((4,4)--(4,4.5)--(3.5,4.5)--(3.5,4)--cycle,black+linewidth(1));
draw((6.8284,5.1716)--(7.0784,5.4216)--(6.8284,5.6716)--(6.5784,5.4216)--cycle,black+linewidth(1));
draw((9.6569,4)--(10.1569,4)--(10.1569,4.5)--(9.6569,4.5)--cycle,black+linewidth(1));
draw((13.6569,4)--(13.1569,4)--(13.1569,3.5)--(13.6569,3.5)--cycle,black+linewidth(1));
draw((13.6569,0)--(13.1569,0)--(13.1569,0.5)--(13.6569,0.5)--cycle,black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(0,4),NW);
label("$C$",(4,4),S);
label("$D$",(4,8),N);
label("$E$",(6.8284,5.1716),S);
label("$F$",(9.6569,8),N);
label("$G$",(9.6569,4),S);
label("$H$",(13.6569,4),NE);
label("$K$",(13.6569,0),E);
[/asy]

Determine the length of $DF.$

[asy]
draw((0,0)--(2.8284,-2.8284)--(5.6568,0),black+linewidth(1));
draw((0,0)--(5.6568,0),black+linewidth(1)+dashed);
draw((2.8284,-2.8284)--(3.0784,-2.5784)--(2.8284,-2.3284)--(2.5784,-2.5784)--cycle,black+linewidth(1));
label("$D$",(0,0),N);
label("$E$",(2.8284,-2.8284),S);
label("$F$",(5.6568,0),N);
[/asy]

level

Level 5

type

Prealgebra

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Source and history

task_id
5b465e46-fa0a-529a-ba16-0dd44c0fec8d
task_key
prealgebra--test--5b465e46-fa0a-529a-ba16-0dd44c0fec8d
task_revision_id
1
config / split
prealgebra / test

Official source

initial import