2007 AMC 8 Problems/Problem 12

Problem

A unit hexagram is composed of a regular hexagon of side length $1$ and its $6$ equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

$[asy] defaultpen(linewidth(0.7)); draw(polygon(3)); pair D=origin+1*dir(270), E=origin+1*dir(150), F=1*dir(30); draw(D--E--F--cycle); [/asy]$

$\mathrm{(A)}\ 1:1 \qquad \mathrm{(B)}\ 6:5 \qquad \mathrm{(C)}\ 3:2 \qquad \mathrm{(D)}\ 2:1 \qquad \mathrm{(E)}\ 3:1$

Solution

The six equilateral triangular extensions fit perfectly into the hexagon meaning the answer is $\boxed{\textbf{(A) }1:1}$

Solution 2

Split the hexagon into six small equilateral triangles. You will see that the six outer triangles can be folded to the hexagon, so the answer is $\boxed{\textbf{(A) }1:1}.$

Video Solution by OmegaLearn

https://youtu.be/abSgjn4Qs34?t=349

~ pi_is_3.14

2007 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2007 AMC 8 Problems/Problem 12

Contents

Problem

Solution

Solution 2

Video Solution by OmegaLearn

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