Problem 66 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Two straight lines intersect at a point \(O\). Points \(A_{1}\), \(A_{2}, \ldots, A_{n}\) are taken on one line and points \(B_{1}, B_{2}, \ldots\), \(B_{n}\) on the other. If the point \(O\) is not to be used, the number of triangles that can be drawn using these points as vertices, is (A) \(n(n-1)\) (B) \(n(n-1)^{2}\) (C) \(n^{2}(n-1)\) (D) \(n^{2}(n-1)^{2}\)

Step-by-Step Solution

Answer

The answer is (C): \(n^2(n-1)\).

Step 1: Understanding the Scenario

Two intersecting lines create a setup where one line has points \(A_1, A_2, \ldots, A_n\) and the other has points \(B_1, B_2, \ldots, B_n\). We want to find out how many triangles can be formed using these points, but without including the intersection point \(O\).

Step 2: Basic Triangle Formation

A triangle can be formed by selecting any three points such that no two points are on the same line. This means selecting one point from the set \(\{A_1, A_2, \ldots, A_n\}\) and two points from the set \(\{B_1, B_2, \ldots, B_n\}\), or vice versa.

Step 3: Calculating Combinations for One Line

If we select one point from \(\{A_1, A_2, \ldots, A_n\}\), there are \(n\) choices. For two points from \(\{B_1, B_2, \ldots, B_n\}\), there are \(\binom{n}{2} = \frac{n(n-1)}{2}\) combinations.

Step 4: Calculating for the Other Line

Similarly, if we select one point from \(\{B_1, B_2, \ldots, B_n\}\), we again have \(n\) choices. For two points from \(\{A_1, A_2, \ldots, A_n\}\), there are \(\binom{n}{2} = \frac{n(n-1)}{2}\) combinations.

Step 5: Total Number of Triangles

To get the total number of triangles formed without \(O\), we add both scenarios: selecting one point from \(\{A_i\}\) and two from \(\{B_i\}\), and vice versa. This gives:\[ n \times \frac{n(n-1)}{2} + n \times \frac{n(n-1)}{2} = 2 \times \frac{n^2(n-1)}{2} = n^2(n-1) \]

Step 6: Selecting the Correct Answer

Out of the given options, the calculation matches option (C): \(n^2(n-1)\). This is the number of triangles that can be formed using the given points.

Key Concepts

Triangles

Triangles are one of the basic shapes in geometry, formed by connecting three non-collinear points with straight lines. A triangle has three sides, three vertices, and three interior angles. These fundamental properties give triangles a unique place in both theoretical and practical mathematics as they form the basis for much of geometry's principles.
To understand how triangles are formed, consider any three points in a plane. If these points are not all on the same line, they can be joined to form a triangle. The distinctive feature of a triangle is that none of the sides are collinear, meaning that no two points lie along the same straight line.
Key points about triangles:

• Triangles are defined by three non-collinear points.
• The sum of the interior angles is always 180 degrees.
• There are different types of triangles based on side lengths (equilateral, isosceles, scalene) and angle measurements (acute, right, obtuse).

Understanding these fundamental properties of triangles is crucial when analyzing geometrical figures involving triangular formations.

Intersecting Lines

Intersecting lines are central to this problem, as they create points through which triangles can potentially form. Two lines that intersect share a point called the point of intersection, denoted here as point O. This point is common to both lines but is not used in forming triangles in this scenario.
When lines intersect, they create angles around the point of intersection, which can further extend to form triangular areas if additional points are decided upon these lines. However, avoiding the point of intersection ensures that no two points are collinear, enabling triangle formation.
Characteristics of intersecting lines:

• They meet at a single point.
• They form four angles around the intersection point.
• The angles opposite each other around the point of intersection are equal (vertically opposite angles).

In the given context, avoiding the use of point O ensures that the created geometrical figures (triangles) do not become degenerate, meaning that all sides remain distinct and non-collinear.

Combinations

Combinatorics is a branch of mathematics focused on counting, especially in situations where there is a finite number of outcomes. In this exercise, combinations are used to determine how many different triangles can be formed without including the intersection point O.
To solve this, we apply the concept of combinations, which allows us to choose a subset of objects (in this case, points) from a larger set without regard to the order. The formula for combinations of choosing \(r\) objects from \(n\) objects is given by:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]
In the problem context, we're interested in forming triangles, which requires selecting three points. This can happen in two ways: choose one point from one line and two from the other; then repeat by reversing these choices. This calculation is essential for determining the variety of triangles possible from the given arrangement of points.

• From one line, select one point: \(n\) ways.
• From the other, choose two points: \(\binom{n}{2}\) ways, calculated as \(\frac{n(n-1)}{2}\).

Adding these together gives the total number of possible triangles without using the intersection point, showcasing the power of combinatorial operations in mathematical problem-solving.