Difference between revisions of "2012 AMC 10B Problems/Problem 21"

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Drawing the points out, it is possible to have a diagram where <math>b=\sqrt{3}a</math>. It turns out that <math>a,</math> <math>2a,</math> and <math>b</math> could be the lengths of a 30-60-90 triangle, and the other 3 <math>a\text{'s}</math> can be the lengths of an equilateral triangle formed from connecting the dots.
 
Drawing the points out, it is possible to have a diagram where <math>b=\sqrt{3}a</math>. It turns out that <math>a,</math> <math>2a,</math> and <math>b</math> could be the lengths of a 30-60-90 triangle, and the other 3 <math>a\text{'s}</math> can be the lengths of an equilateral triangle formed from connecting the dots.
 
So, <math>b=\sqrt{3}a</math>, so <math>b:a= \sqrt{3}=(A)</math>
 
So, <math>b=\sqrt{3}a</math>, so <math>b:a= \sqrt{3}=(A)</math>
 +
<asy>draw((0, 0)--(1/2, sqrt(3)/2)--(1, 0)--cycle);
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draw((1/2, sqrt(3)/2)--(2, 0)--(1,0));
 +
label("$a$", (0, 0)--(1, 0), S);
 +
label("$a$", (1, 0)--(2, 0), S);
 +
label("$a$", (0, 0)--(1/2, sqrt(3)/2), NW);
 +
label("$a$", (1, 0)--(1/2, sqrt(3)/2), NE);
 +
label("$b=\sqrt{3}a$", (1/2, sqrt(3)/2)--(2, 0), NE);
 +
</asy>
  
 
== See Also ==
 
== See Also ==

Revision as of 14:40, 20 March 2013

Problem 21

Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$, $a$, $a$, $a$, $2a$, and $b$. What is the ratio of $b$ to $a$?

$\textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi$

Solution

When you see that there are lengths a and 2a, one could think of 30-60-90 triangles. Since all of the other's lengths are a, you could think that $b=\sqrt{3}a$. Drawing the points out, it is possible to have a diagram where $b=\sqrt{3}a$. It turns out that $a,$ $2a,$ and $b$ could be the lengths of a 30-60-90 triangle, and the other 3 $a\text{'s}$ can be the lengths of an equilateral triangle formed from connecting the dots. So, $b=\sqrt{3}a$, so $b:a= \sqrt{3}=(A)$ [asy]draw((0, 0)--(1/2, sqrt(3)/2)--(1, 0)--cycle); draw((1/2, sqrt(3)/2)--(2, 0)--(1,0)); label("$a$", (0, 0)--(1, 0), S); label("$a$", (1, 0)--(2, 0), S); label("$a$", (0, 0)--(1/2, sqrt(3)/2), NW); label("$a$", (1, 0)--(1/2, sqrt(3)/2), NE); label("$b=\sqrt{3}a$", (1/2, sqrt(3)/2)--(2, 0), NE); [/asy]

See Also

2012 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
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