Problem 70
Question
There are 10 points in a plane, out of these 6 are collinear. If \(N\) is the
number of triangles formed by joining these points. Then:
(a) \(N \leq 100\)
(b) \(100
Step-by-Step Solution
Verified Answer
The number of triangles formed is (a) \(N \leq 100\).
1Step 1: Calculate Total Possible Triangles
The total number of triangles that can be formed with 10 points is calculated by choosing any 3 points out of the 10.Use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here, \( n = 10 \) and \( r = 3 \).\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \] Thus, the total possible triangles is 120.
2Step 2: Subtract Collinear Point Triangles
Among the 10 points, 6 are collinear. No triangle can be formed with collinear points. Calculate the number of ways to choose 3 points from these 6, which would form straight lines (not triangles):\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]These 20 configurations are not valid triangles.
3Step 3: Calculate Valid Triangles
Subtract the invalid formations (20 collinear 'triangles') from the total possible triangles (120):\[ 120 - 20 = 100 \]Thus, the number of valid triangles formed is 100.
Key Concepts
GeometryTrianglesCollinearity
Geometry
Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and shapes. It involves understanding how different shapes fit together and interact in space. The problem in question uses geometry to explore the formation of triangles from a set of points on a plane. In geometry, **points** are fundamental objects that don't have a size or dimension, but they serve as precise locations in space. When points are connected, they form lines, angles, and shapes.
In two-dimensional geometry, common shapes include circles, squares, rectangles, and notably for our problem, triangles. By understanding how these shapes are formed and how they relate to one another, we can solve complex spatial problems. Concepts like parallel lines, congruence, and similarity are key aspects of this study. In the case of our problem, understanding lines and configurations of points helps in solving the problem of how many triangles are possible given certain conditions.
In two-dimensional geometry, common shapes include circles, squares, rectangles, and notably for our problem, triangles. By understanding how these shapes are formed and how they relate to one another, we can solve complex spatial problems. Concepts like parallel lines, congruence, and similarity are key aspects of this study. In the case of our problem, understanding lines and configurations of points helps in solving the problem of how many triangles are possible given certain conditions.
Triangles
Triangles are one of the simplest and most fundamental shapes in geometry. Defined by three sides and three angles, triangles are the result of connecting three non-collinear points. Each of these connections forms a **side** of the triangle. The sum of the internal angles of a triangle always equals 180 degrees.
There are several types of triangles based on their angles and sides:
Understanding triangle formation is essential for solving problems in combinatorics that involve choosing combinations of points to form valid triangles.
There are several types of triangles based on their angles and sides:
- **Equilateral Triangle**: All sides and angles are equal.
- **Isosceles Triangle**: Two sides are of equal length, and two angles are identical.
- **Scalene Triangle**: All sides and angles are different.
Understanding triangle formation is essential for solving problems in combinatorics that involve choosing combinations of points to form valid triangles.
Collinearity
Collinearity is a geometric property where three or more points lie on a single straight line. In the context of combinatorics and this particular problem, collinear points pose unique challenges.
When you have a set of points, like in this problem where six out of ten points are collinear, those collinear points cannot form a triangle. If you attempt to connect three collinear points, instead of forming a triangle, they form a segment of a straight line. Therefore, in our exercise, we must subtract any potential 'triangles' formed by collinear point triplets from the total number of triangles.
The ability to identify and account for collinearity ensures precise calculations in combinatorics problems. Recognizing collinear objects simplifies many geometrical computations and helps accurately address exercises like this where geometry and combinatorics intersect.
When you have a set of points, like in this problem where six out of ten points are collinear, those collinear points cannot form a triangle. If you attempt to connect three collinear points, instead of forming a triangle, they form a segment of a straight line. Therefore, in our exercise, we must subtract any potential 'triangles' formed by collinear point triplets from the total number of triangles.
The ability to identify and account for collinearity ensures precise calculations in combinatorics problems. Recognizing collinear objects simplifies many geometrical computations and helps accurately address exercises like this where geometry and combinatorics intersect.
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