Formation of Geometrical Figures
⦿ Number of straight lines from given n points:
Given, there are ‘n’ points in a plain. no three of them are collinear, then number of straight lines is same as number of ways of selecting 2 points out of ‘n’ points = ⁿC₂
Example:- There are 10 points in a plane. No three of them are collinear. Then how many straight lines can be drawn from these points.
Solution:- ¹⁰C₂ Answer
⦿ Number of straight lines from given ‘n’ points, out of them exactly ‘k’ points are collinear Concept
if no three points would have been collinear then total number of straight lines = ⁿC₂
if out of ‘k’ points no three points would have been collinear then number of straight lines through these ‘k’ points = ᵏC₂
But since these ‘k’ points are collinear so these ‘k’ point will not form ᵏC₂ lines but only 1 line can be made from these ‘k’ collinear points. But in ⁿC₂ calculation these ᵏC₂ lines are also included
Example:- There are 10 points in a plane. Out of these 4 are collinear. Then how many straight lines can be drawn from these 10 points?
Solution:-
¹⁰C₂ – ⁴C₂ + 1
= 45 – 6 + 1
= 40 Answer
⦿ Number of Triangle from given ‘n’ points is no three of them are collinear = ⁿC₃
Example:- How many triangle can be drawn from 10 points in a plane if no three of them are collinear?
Solution:-
= ¹⁰C₃
=
= 120 Answer
⦿ Number of polygon of ‘r’ sides from given ‘n’ points if no three of them are collinear = ⁿCᵣ
Proof:- A polygon of ‘r’ sides has ‘r’ vertices.
∴ Number of total polygon = number of ways of selecting ‘r’ things out of ‘n’ things
= ⁿCᵣ
Example:- How many octagon can be drawn from 10 points in a plane if no three of them are collinear?
Solution:- ¹⁰C₈
=
= 45 Answer
⦿ Number of diagonal of a polygon of n sides
Since diagonal is formed by joining two vertices i.e. by joining two points . Since polygon of n sides has n vertices. Hence total number of straight lines that can be drawn out of these ‘n’ points
Hence number of diagonal = ⁿC₂ – number of sides
= ⁿC₂ – n
= – n
Example:- What is the number of diagonal of a hexagon?
Solution:- ⁶C₂ – 6
= 9 Answer
Example:- If a polygon has 35 diagonal. Then how many sides that polygon has?
Solution:-
= 35
n² – 3n – 70 = 0
n² -10n + 7n – 70 = 0
n(n-10) + 7(n-10) = 0
(n-10)(n+7) = 0
n=10
∴ polygon has 10 sides Answer