Difference between revisions of "2007 iTest Problems/Problem 15"

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== Problem ==
 
== Problem ==
  
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1 and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one vertex of the equilateral triangle, and exactly two vertices of the square. What is the radius of the circle?
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Form a pentagon by taking a square of side length <math>1</math> and an equilateral triangle of side length <math>1</math> and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one vertex of the equilateral triangle, and exactly two vertices of the square. What is the radius of the circle?
  
<math>\textbf{(A) }\dfrac23\hspace{14.4em}
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<math>\textbf{(A) }\dfrac23\qquad
\textbf{(B) }\dfrac34\hspace{14.4em}
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\textbf{(B) }\dfrac34\qquad
\textbf{(C) }1
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\textbf{(C) }1\qquad
\textbf{(D) }\dfrac54\hspace{14.4em}
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\textbf{(D) }\dfrac54\qquad
\textbf{(E) }\dfrac43\hspace{14.4em}
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\textbf{(E) }\dfrac43\qquad \\ </math>
\textbf{(F) }\dfrac{\sqrt2}2
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<math>\textbf{(F) }\dfrac{\sqrt2}2\qquad
\textbf{(G) }\dfrac{\sqrt3}2\hspace{13.5em}\&#036;
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\textbf{(G) }\dfrac{\sqrt3}2\qquad
</math>\textbf{(H) }\sqrt2\hspace{13.8em}
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\textbf{(H) }\sqrt2\qquad
\textbf{(I) }\sqrt3
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\textbf{(I) }\sqrt3\qquad
\textbf{(J) }\dfrac{1+\sqrt3}2\hspace{12em}
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\textbf{(J) }\dfrac{1+\sqrt3}2\qquad \\ </math>
\textbf{(K) }\dfrac{2+\sqrt6}2\hspace{11.9em}
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<math>\textbf{(K) }\dfrac{2+\sqrt6}2\qquad
\textbf{(L) }\dfrac76
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\textbf{(L) }\dfrac76\qquad
\textbf{(M) }\dfrac{2+\sqrt6}4\hspace{11.5em}
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\textbf{(M) }\dfrac{2+\sqrt6}4\qquad
\textbf{(N) }\dfrac45\hspace{14.4em}
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\textbf{(N) }\dfrac45\qquad
\textbf{(O) }2007 $
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\textbf{(O) }2007\qquad </math>
  
 
== Solution ==
 
== Solution ==

Revision as of 22:18, 7 October 2014

Problem

Form a pentagon by taking a square of side length $1$ and an equilateral triangle of side length $1$ and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one vertex of the equilateral triangle, and exactly two vertices of the square. What is the radius of the circle?

$\textbf{(A) }\dfrac23\qquad \textbf{(B) }\dfrac34\qquad \textbf{(C) }1\qquad \textbf{(D) }\dfrac54\qquad \textbf{(E) }\dfrac43\qquad \$ (Error compiling LaTeX. ! Missing $ inserted.) $\textbf{(F) }\dfrac{\sqrt2}2\qquad \textbf{(G) }\dfrac{\sqrt3}2\qquad \textbf{(H) }\sqrt2\qquad \textbf{(I) }\sqrt3\qquad \textbf{(J) }\dfrac{1+\sqrt3}2\qquad \$ (Error compiling LaTeX. ! Missing $ inserted.) $\textbf{(K) }\dfrac{2+\sqrt6}2\qquad \textbf{(L) }\dfrac76\qquad \textbf{(M) }\dfrac{2+\sqrt6}4\qquad \textbf{(N) }\dfrac45\qquad \textbf{(O) }2007\qquad$

Solution

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