Geometry Concepts and Questions : Number of Triangles or Circles Formed from n Points – Must Practice Set for CAT, XAT & Other MBA Exams
May 29, 2024 2025-05-29 16:55Geometry Concepts and Questions : Number of Triangles or Circles Formed from n Points – Must Practice Set for CAT, XAT & Other MBA Exams
Geometry Concepts and Questions : Number of Triangles or Circles Formed from n Points – Must Practice Set for CAT, XAT & Other MBA Exams
Amiya Kumar 🔺 ⭕Number of Triangles or Circles Formed from n Points – Geometry Concept
In geometry, a triangle is formed by choosing 3 non-collinear points. The concept of how many Triangles or Circles can be formed from \( n \) points is an important topic for exams like CAT, XAT, SSC CGL, NTSE, and other aptitude tests.
1. When No Three Points Are Collinear
If no three points among \( n \) are collinear, then the number of Triangles or Circles is simply:
Each set of 3 distinct points uniquely determines a triangle.
2. When m Points Are Collinear
If \( m \) out of \( n \) points are collinear, they lie on the same straight line and cannot form a triangle among themselves.
Hence, total Triangles or Circles = Total possible Triangles or Circles − Invalid Triangles or Circles from collinear set:
3. General Case: Multiple Collinear Groups
If multiple groups of collinear points exist (say sets of sizes \( m_1, m_2, \dots \)), then:
📌 Examples
Example 1: Out of 10 points, no three are collinear. Then:
Example 2: Out of 10 points, 4 are collinear. Then:
Example 3: Out of 12 points, two collinear groups of 4 and 3 points. Then:
🔺 Practice Questions – Triangles Formed from n Points
Click below to reveal answers and step-by-step explanations.
1) Out of 10 points on a plane, 8 are collinear. How many triangles can be formed?
🔽 Show AnswerAnswer: 64
Solution: \( \binom{10}{3} - \binom{8}{3} = 120 - 56 = 64 \)
2) Out of 8 points on a plane, 6 are collinear. How many triangles can be formed?
🔽 Show AnswerAnswer: 36
Solution: \( \binom{8}{3} - \binom{6}{3} = 56 - 20 = 36 \)
3) If no three of the 14 points on a plane are collinear, how many triangles can be formed?
🔽 Show AnswerAnswer: 364
Solution: \( \binom{14}{3} = \frac{14 \cdot 13 \cdot 12}{6} = 364 \)
4) If no three of the 9 points on a plane are collinear, how many triangles can be formed?
🔽 Show AnswerAnswer: 84
Solution: \( \binom{9}{3} = \frac{9 \cdot 8 \cdot 7}{6} = 84 \)
5) Out of 7 points on a plane, 3 are collinear. How many triangles can be formed?
🔽 Show AnswerAnswer: 34
Solution: \( \binom{7}{3} - \binom{3}{3} = 35 - 1 = 34 \)
6) Out of 12 points on a plane, 5 are collinear. How many triangles can be formed?
🔽 Show AnswerAnswer: 206
Solution: \( \binom{12}{3} - \binom{5}{3} = 220 - 10 = 210 \)
7) If all 8 points lie on the same line, how many triangles can be formed?
🔽 Show AnswerAnswer: 0
Solution: All points collinear → no triangle possible.
8) Out of 9 points, 3 are collinear. How many triangles can be formed?
🔽 Show AnswerAnswer: 83
Solution: \( \binom{9}{3} - \binom{3}{3} = 84 - 1 = 83 \)
9) From 11 points on a plane, 4 are collinear. How many triangles can be formed?
🔽 Show AnswerAnswer: 161
Solution: \( \binom{11}{3} - \binom{4}{3} = 165 - 4 = 161 \)
10) Out of 15 points, no three are collinear. How many triangles can be formed?
🔽 Show AnswerAnswer: 455
Solution: \( \binom{15}{3} = \frac{15 \cdot 14 \cdot 13}{6} = 455 \)
