Consider a sided polygon, then, the probability that if three vertices are selected at random, the triangle so formed will contain the centre of the polygon is the value of  is

# Consider a sided polygon, then, the probability that if three vertices are selected at random, the triangle so formed will contain the centre of the polygon is the value of  is

1. A
5
2. B
3
3. C
4
4. D
9

Fill Out the Form for Expert Academic Guidance!l

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)

### Solution:

Now, let us assume that one vertex of the required triangle as .
Now, let us find the number of triangles by considering the number of pairs of remaining sides of the triangle.
Let us assume that the remaining vertices of the triangle as where,   represents the vertex number.
To find the number of triangles that contain the center of the polygon.
We know that for a sided polygon the gap between the other two vertices must be at most , that is the gap should be or less than in order to get the center of the polygon in the triangle.
Here, we considered the first vertex as .
By using the above condition the possible vertices of   are and .
Now, let us take the possibilities of b for each vertex of .
(1)  For the possible vertex of is .
(2)  For  the possible vertex of is or .
(3)  For  the possible vertex of b is or or .
(4)  For  the possible vertex of b is or or or .
Here, we can see that there are total of triangles that contain the center of the polygon in the triangle.
Let us assume that the number of total number of outcomes as then we get,
Let us assume that the total number of outcomes as .
Here, we can see that we have already selected one vertex as then the number of ways of selecting the remaining vertices from  vertices is given as
We know that the formula of probability a
By using the formula we get the required probability as
We are given that the required probability as .
By comparing the two probabilities we get .
Hence, Option (1) is the correct answer.

## Related content

 Area of Square Area of Isosceles Triangle Pythagoras Theorem Triangle Formula Perimeter of Triangle Formula Area Formulae Volume of Cone Formula Matrices and Determinants_mathematics Critical Points Solved Examples Type of relations_mathematics

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)