Q. 215.0( 1 Vote )

Find the number of (i) diagonals (ii) triangles formed in a decagon.

Answer :

i. We know that the decagon has 10 vertices and each side and diagonal can be formed by joining two vertices of a hexagon,


We know that decagon has 10 sides,


Let us assume the no. of diagonals of the hexagon are N,


N = (no. of lines formed on joining any two vertices) – (no. of sides of the hexagon)


N = 10C2 – 10


We know that ,


And also n! = (n)(n – 1)......2.1





N = 45 – 10


N = 35


The total no. of diagonals formed is 35.


ii. Given that we need to find the no. of triangles that can be drawn in a decagon.


We know that 3 points are required to draw a triangle.


We know that decagon has 10 sides


Let us assume the no. of triangles formed be N1,


N1 = (total no. of triangles formed by all 10 points)


N1 = 10C3


We know that ,


And also n! = (n)(n – 1)......2.1





N1 = 120


The total no. of ways of different lines formed are 120.


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