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MATH / All of the triangles in the figure and the central hexagon are equilateral.…

Problem

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

problem

All of the triangles in the figure and the central hexagon are equilateral. Given that AC¯\overline{AC} is 3 units long, how many square units, expressed in simplest radical form, are in the area of the entire star? [asy] import olympiad; import geometry; import graph; size(150); defaultpen(linewidth(0.8)); pair[] vertices; for(int i = 0; i < 6; ++i){ vertices[i] = dir(30 + 60*i); } draw(vertices[0]--vertices[2]--vertices[4]--cycle); draw(vertices[1]--vertices[3]--vertices[5]--cycle); label("DD",vertices[0],NE); label("CC",vertices[1],N); label("BB",vertices[2],NW); label("AA",vertices[3],SW); label("FF",vertices[4],S); label("EE",vertices[5],SE); [/asy]
Plain-text mathematical notation (without MathML)
All of the triangles in the figure and the central hexagon are equilateral. Given that (AC)¯ is 3 units long, how many square units, expressed in simplest radical form, are in the area of the entire star? [asy]
import olympiad; import geometry; import graph; size(150); defaultpen(linewidth(0.8));
pair[] vertices;
for(int i = 0; i < 6; ++i){

vertices[i] = dir(30 + 60*i);
}
draw(vertices[0]--vertices[2]--vertices[4]--cycle);
draw(vertices[1]--vertices[3]--vertices[5]--cycle);
label("D",vertices[0],NE); label("C",vertices[1],N); label("B",vertices[2],NW); label("A",vertices[3],SW);
label("F",vertices[4],S); label("E",vertices[5],SE);
[/asy]
Original LaTeX notation
All of the triangles in the figure and the central hexagon are equilateral. Given that $\overline{AC}$ is 3 units long, how many square units, expressed in simplest radical form, are in the area of the entire star? [asy]
import olympiad; import geometry; import graph; size(150); defaultpen(linewidth(0.8));
pair[] vertices;
for(int i = 0; i < 6; ++i){

vertices[i] = dir(30 + 60*i);
}
draw(vertices[0]--vertices[2]--vertices[4]--cycle);
draw(vertices[1]--vertices[3]--vertices[5]--cycle);
label("$D$",vertices[0],NE); label("$C$",vertices[1],N); label("$B$",vertices[2],NW); label("$A$",vertices[3],SW);
label("$F$",vertices[4],S); label("$E$",vertices[5],SE);
[/asy]

level

Level 4

type

Geometry

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Source and history

task_id
9e0841e6-375b-5a57-b089-89cd49a56d52
task_key
geometry--test--9e0841e6-375b-5a57-b089-89cd49a56d52
task_revision_id
1
config / split
geometry / test

Official source

initial import