Difference between revisions of "2012 AMC 10B Problems/Problem 21"
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− | When you see a and 2a, | + | ==Problem 21== |
− | Drawing the points out, it is possible to have a diagram where <math>b=\sqrt{3a}</math> | + | Four distinct points are arranged on a plane so that the segments connecting them have lengths <math>a</math>, <math>a</math>, <math>a</math>, <math>a</math>, <math>2a</math>, and <math>b</math>. What is the ratio of <math>b</math> to <math>a</math>? |
+ | |||
+ | <math>\textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi</math> | ||
+ | |||
+ | ==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 <math>b=\sqrt{3a}</math>. | ||
+ | Drawing the points out, it is possible to have a diagram where <math>b=\sqrt{3a}</math>. It turns out that <math>a, 2a, and b</math> could be the lengths of a 30-60-90 triangle, and the other 3 <math>"a's"</math> can be the lengths of an equilateral triangle formed from connecting the dots. | ||
So, <math>b=\sqrt{3a}</math>, so <math>b:a= \sqrt{3}=(A)</math> | So, <math>b=\sqrt{3a}</math>, so <math>b:a= \sqrt{3}=(A)</math> |
Revision as of 00:05, 14 March 2012
Problem 21
Four distinct points are arranged on a plane so that the segments connecting them have lengths , , , , , and . What is the ratio of to ?
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 . Drawing the points out, it is possible to have a diagram where . It turns out that could be the lengths of a 30-60-90 triangle, and the other 3 can be the lengths of an equilateral triangle formed from connecting the dots. So, , so