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MATH / Points E and F are located on square ABCD so that △BEF is equilateral. What is the ratio of the area of △DEF to that of △ABE? [asy] pair A,B,C,D,I,F; A=(0,0); B=(10,0); C=(10,10); …

Problem

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

problem

Points EE and FF are located on square ABCDABCD so that △BEF\triangle BEF is equilateral. What is the ratio of the area of △DEF\triangle DEF to that of △ABE\triangle ABE? [asy] pair A,B,C,D,I,F; A=(0,0); B=(10,0); C=(10,10); D=(0,10); F=(7.4,10); I=(0,2.6); draw(B--I--F--cycle,linewidth(0.7)); draw(A--B--C--D--cycle,linewidth(0.7)); label("AA",A,S); label("BB",B,S); label("CC",C,N); label("DD",D,N); label("EE",I,W); label("FF",F,N); [/asy]
Plain-text mathematical notation (without MathML)
Points E and F are located on square ABCD so that △BEF is equilateral. What is the ratio of the area of △DEF to that of △ABE?

[asy]
pair A,B,C,D,I,F;
A=(0,0);
B=(10,0);
C=(10,10);
D=(0,10);
F=(7.4,10);
I=(0,2.6);
draw(B--I--F--cycle,linewidth(0.7));
draw(A--B--C--D--cycle,linewidth(0.7));
label("A",A,S);
label("B",B,S);
label("C",C,N);
label("D",D,N);
label("E",I,W);
label("F",F,N);
[/asy]
Original LaTeX notation
Points $E$ and $F$ are located on square $ABCD$ so that $\triangle BEF$ is equilateral. What is the ratio of the area of $\triangle DEF$ to that of $\triangle ABE$?

[asy]
pair A,B,C,D,I,F;
A=(0,0);
B=(10,0);
C=(10,10);
D=(0,10);
F=(7.4,10);
I=(0,2.6);
draw(B--I--F--cycle,linewidth(0.7));
draw(A--B--C--D--cycle,linewidth(0.7));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$D$",D,N);
label("$E$",I,W);
label("$F$",F,N);
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
fd774e6e-0761-5220-95b2-443eb18fd47c
task_key
geometry--test--fd774e6e-0761-5220-95b2-443eb18fd47c
task_revision_id
1
config / split
geometry / test

Official source

initial import