Some points are plotted on a plane. Each point is joined with all rema

1.	Some points are plotted on a plane. Each point is joined with all remaining points by line segments. Find the number of points if the number of line segments are 10.
Answer» Number of distinct line segments that can be formed out of n-points = \(\frac{n(n-1)}{2}\) Given: No. of line segments \(\frac{n(n-1)}{2}\) = 10 ⇒ n² – n = 20 ⇒ n² – n – 20 = 0 ⇒ n² – 5n + 4n – 20 = 0 ⇒ n(n – 5) + 4(n – 5) = 0 ⇒ (n – 5) (n + 4) = 0 ⇒ n – 5 = 0 (or) n + 4 = 0 ⇒ n = 5 (or) -4 ∴ n = 5 [n – can’t be negative]

Some points are plotted on a plane. Each point is joined with all remaining points by line segments. Find the number of points if the number of line segments are 10.

Answer»

Number of distinct line segments that can be formed out of n-points = \(\frac{n(n-1)}{2}\)

Given: No. of line segments

\(\frac{n(n-1)}{2}\) = 10

⇒ n² – n = 20

⇒ n² – n – 20 = 0

⇒ n² – 5n + 4n – 20 = 0

⇒ n(n – 5) + 4(n – 5) = 0

⇒ (n – 5) (n + 4) = 0

⇒ n – 5 = 0 (or) n + 4 = 0

⇒ n = 5 (or) -4

∴ n = 5 [n – can’t be negative]