The number of diagonals of a polygon is 35. If A, B are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having AB as one of its sides is:
The number of diagonals of a polygon is 35. If A, B are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having AB as one of its sides is:
1
8
10
12
The Correct Option is C
Solution and Explanation
The correct option is: (C) 10.
Number of diagonals=2n(n−3)
Given that the number of diagonals is 35, we can set up an equation and solve for n:
n(n−3)=70
Now, we need to find two integers n and n−3 whose product is 70. The pairs of integers that satisfy this condition are (n,n−3)=(10,7) and (−3)=(35,32)(n,n−3)=(35,32).
However, in the context of a polygon, the number of sides cannot be negative or zero, so we discard the solution(n,n−3)=(35,32).
Therefore, the number of sides of the polygon is n=10.
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Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
