Challenge Problem: 9/25/06

[brien]

I follow it up to the evaluation of Statement 2. Isn't putting in 360 as the sum of the exterior angles a carryover from statement 1? Won't that change the answer from B to C?

http://www.manhattangmat.com/ChallengeP ... hallID=263

Question
A certain game board is in the shape of a non-convex polygon, with spokes that extend from each vertex to the center of the board. If each spoke is 8 inches long, and spokes are used nowhere else on the board, what is the sum of the interior angles of the polygon?

(1) The sum of the exterior angles of the polygon is 360º.

(2) The sum of the exterior angles is equal to five times the total length of all of the spokes used.

ANSWER
The formula for the sum of the interior angles of a non-convex polygon is (n - 2)(180), where n represents the number of sides. To find the sum of the interior angles of the polygon then, we need to know the number of sides. We can therefore rephrase the question:

How many sides does the game board have?

(1) INSUFFICIENT: It tells us nothing about the number of sides. The sum of the exterior angles for any non-convex polygon is 360.

(2) SUFFICIENT: The sum of the exterior angles = 5 × length of each spoke × number of spokes.

360 = 5(8)(x)
360 = 40x
9 = x

The game board has nine sides. The sum of its interior angles is (9 - 2)(180) = 1260.

The correct answer is B.

[rfernandez]

The 360 in the evaluation of (2) did not come from (1). This is a property of all polygons: the exterior angles add up to 360.