Triangle ABC is inscribed in a circle, and ∠ B=∠ C=4∠ A. If B and C are adjacent vertices of a regular polygon of n sides inscribed in this circle, then n=

1994 AMC 12 Problem 18

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18.

Triangle

ABC

is inscribed in a circle, and

\angle B=\angle C=4\angle A.

B

and

C

are adjacent vertices of a regular polygon of

n

sides inscribed in this circle, then

n=

5

7

9

15

18

Use the triangle angle sum to find

\angle A

The central angle subtending

\overline{BC}

is twice

\angle A

The angle sum gives

9\angle A=180^\circ,

\angle A=20^\circ.

The central angle subtending chord

\overline{BC}

is therefore

40^\circ.

Adjacent vertices of a regular

n

-gon subtend

\frac{360^\circ}{n},

\frac{360^\circ}{n}=40^\circ,

and

n=9.

Thus the correct answer is C.