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MATH / Equilateral triangle DEF is inscribed in equilateral triangle ABC as shown with (ED)¯⊥(BC)¯. What is the ratio of the area of △DEF to the area of △ABC? [asy] import olympiad; impor…

Problem

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

problem

Equilateral triangle DEFDEF is inscribed in equilateral triangle ABCABC as shown with ED¯⊥BC¯.\overline{ED}\perp\overline{BC}. What is the ratio of the area of △DEF\triangle DEF to the area of △ABC\triangle ABC? [asy] import olympiad; import geometry; size(100); defaultpen(linewidth(0.8)); draw(dir(90)--dir(210)--(dir(-30))--cycle); pair[] inters = intersectionpoints(Circle(origin,1/sqrt(3)),dir(90)--dir(210)--(dir(-30))--cycle); inters[5] = dir(210) + ((dir(-30) - dir(210))/3); draw(inters[0]--inters[2]--inters[5]--cycle); draw(rightanglemark(inters[2],inters[5],dir(210),4)); dot("AA",dir(90),N); dot("CC",dir(210),W); dot("BB",dir(-30),E); dot("FF",inters[0],E); dot("EE",inters[2],W); dot("DD",inters[5],S); [/asy]
Plain-text mathematical notation (without MathML)
Equilateral triangle DEF is inscribed in equilateral triangle ABC as shown with (ED)¯⊥(BC)¯. What is the ratio of the area of △DEF to the area of △ABC? [asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
draw(dir(90)--dir(210)--(dir(-30))--cycle);
pair[] inters = intersectionpoints(Circle(origin,1/sqrt(3)),dir(90)--dir(210)--(dir(-30))--cycle);
inters[5] = dir(210) + ((dir(-30) - dir(210))/3);
draw(inters[0]--inters[2]--inters[5]--cycle);
draw(rightanglemark(inters[2],inters[5],dir(210),4));
dot("A",dir(90),N);
dot("C",dir(210),W);
dot("B",dir(-30),E);
dot("F",inters[0],E);
dot("E",inters[2],W);
dot("D",inters[5],S);
[/asy]
Original LaTeX notation
Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ as shown with $\overline{ED}\perp\overline{BC}.$ What is the ratio of the area of $\triangle DEF$ to the area of $\triangle
ABC$? [asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
draw(dir(90)--dir(210)--(dir(-30))--cycle);
pair[] inters = intersectionpoints(Circle(origin,1/sqrt(3)),dir(90)--dir(210)--(dir(-30))--cycle);
inters[5] = dir(210) + ((dir(-30) - dir(210))/3);
draw(inters[0]--inters[2]--inters[5]--cycle);
draw(rightanglemark(inters[2],inters[5],dir(210),4));
dot("$A$",dir(90),N);
dot("$C$",dir(210),W);
dot("$B$",dir(-30),E);
dot("$F$",inters[0],E);
dot("$E$",inters[2],W);
dot("$D$",inters[5],S);
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
a56d770f-4148-567e-834a-d208cb18a722
task_key
geometry--test--a56d770f-4148-567e-834a-d208cb18a722
task_revision_id
1
config / split
geometry / test

Official source

initial import