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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

Answer published by the source. Consult the official source to check your work against its answer.

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] If perpendicular EMEM is drawn from EE to DFDF (a close-up is shown below), and if xx is the length of EM,EM, then what is $x^2?$ [asy] draw((0,0)--(2.8284,-2.8284)--(5.6568,0)--cycle,black+linewidth(1)); draw((2.8284,0)--(2.8284,-2.8284),black+linewidth(1)+dashed); draw((2.8284,0)--(3.0784,0)--(3.0784,-0.25)--(2.8284,-0.25)--cycle,black+linewidth(1)); label("DD",(0,0),N); label("EE",(2.8284,-2.8284),S); label("FF",(5.6568,0),N); label("MM",(2.8284,0),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]

If perpendicular EM is drawn from E to DF (a close-up is shown below), and if x is the length of EM, then what is $x^2?$

[asy]
draw((0,0)--(2.8284,-2.8284)--(5.6568,0)--cycle,black+linewidth(1));
draw((2.8284,0)--(2.8284,-2.8284),black+linewidth(1)+dashed);
draw((2.8284,0)--(3.0784,0)--(3.0784,-0.25)--(2.8284,-0.25)--cycle,black+linewidth(1));
label("D",(0,0),N);
label("E",(2.8284,-2.8284),S);
label("F",(5.6568,0),N);
label("M",(2.8284,0),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]

If perpendicular $EM$ is drawn from $E$ to $DF$ (a close-up is shown below), and if $x$ is the length of $EM,$ then what is $x^2?$

[asy]
draw((0,0)--(2.8284,-2.8284)--(5.6568,0)--cycle,black+linewidth(1));
draw((2.8284,0)--(2.8284,-2.8284),black+linewidth(1)+dashed);
draw((2.8284,0)--(3.0784,0)--(3.0784,-0.25)--(2.8284,-0.25)--cycle,black+linewidth(1));
label("$D$",(0,0),N);
label("$E$",(2.8284,-2.8284),S);
label("$F$",(5.6568,0),N);
label("$M$",(2.8284,0),N);
[/asy]

level

Level 5

type

Prealgebra

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Source and history

task_id
3e0160ad-1cf6-57e4-8d77-5aac8369f35c
task_key
prealgebra--test--3e0160ad-1cf6-57e4-8d77-5aac8369f35c
task_revision_id
1
config / split
prealgebra / test

Official source

initial import