Use the letter n for the number of sides. How would you represent the number of

Question

Use the letter n for the number of sides. How would you represent the number of diagonals using n? n-3 How would you represent the number of triangles formed, using n? n Let n represents the total number of sides in a given n(n-3) 4. n-sided polygon, n-3 represents the number of diagonals 2 that can be drawn from one vertex. n(n-3) gives a number of diagonals that would overly represent the total number of diagonals in the polygon. Why is this expression divided by 2? What is the result of this product? 180(n-2)???????????????

Explanation / Answer

4.

Let the number of vertices of a polygon be n

1 vertex has (n - 3) diagonals, since we have to leave out the 2 neighbouring point and the point itself.

We have n vertices so there should be n(n - 3) diagonals.

But the problem here is this, we counting both the diagonal from going out from a vertex and also the one coming back which are the same.

For Example, a diagonal is counted from vertex A to B. Now the same should not be counted from B to A.

Hence we divide n(n - 3) by 2.

We should know that sum of the angles in a triangle is 180.

Now, lets find sum of interior angles of a polygon.

the number of triangles formed by vertices of an n sided polygon is (n - 2). For example, if a pentagon has 5 sides, then we can form 3 triangles [ Note that a vetex should not be connected to its adjacent vertices]

So, the sum of interior angles = No. of triangles . x Sum of angles in a triangle

= (n - 2)180

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