AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 12 Medium

If three vertices are chosen at random from a regular decagon, the probability that the chosen three points form a right triangle is （   ）．

  • A.

    \frac{1}{4}

  • B.

    \frac{1}{3}

  • C.

    \frac{1}{2}

  • D.

    \frac{2}{3}

  • E.

    \frac{3}{4}

Answer:B

 

 

From the 10 vertices of a regular decagon, choosing any 3 points gives _{10}C_{3} = 120 possible selections.

 

Consider the regular decagon as inscribed in a circle. For the 3 chosen points to form a right triangle, the side opposite the right angle must be a diameter.

 

There are _{5}C_{1} = 5 ways to choose a diameter of the circle, and for each diameter, there are _{8}C_{1} = 8 choices for the right-angle vertex.

 

Thus, the probability that the chosen 3 points form a right triangle is

\frac{_{5}C_{1}\times _{8}C_{1}}{_{10}C_{3}} = \frac{5 \times 8}{120} = \tfrac{1}{3}.

 

Therefore, the answer is B.

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