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A polygon has a sum of interior angles equal to 1980 degrees. How many triangles can it be subdivided into, and how many sides does it have?

Mathematics

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Here's how to solve the problem:

a) The number of triangles the polygon can be subdivided into.

  1. Recall the formula for the sum of interior angles of a polygon.

    The sum of the interior angles of a polygon with nn sides is given by S=(n−2)×180∘S = (n-2) \times 180^\circ. The term (n−2)(n-2) represents the number of non-overlapping triangles the polygon can be divided into.

  2. Use the given sum to find the number of triangles.

    Given that the sum of the interior angles is 1980∘1980^\circ. Let TT be the number of triangles.

    T×180∘=1980∘T \times 180^\circ = 1980^\circ T=1980∘180∘T = \frac{1980^\circ}{180^\circ} T=11T = 11

    The number of triangles the polygon can be subdivided into is 11.

    b) The number of sides the polygon has.

  3. Relate the number of triangles to the number of sides.

    The number of triangles a polygon can be subdivided into is n−2n-2, where nn is the number of sides.

    n−2=Tn-2 = T n−2=11n-2 = 11 n=11+2n = 11 + 2 n=13n = 13

    The number of sides the polygon has is 13.

    The final answers are:

    a) The number of triangles the polygon can be subdivided into is 11\boxed{11}.

    b) The number of sides the polygon has is 13\boxed{13}.

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