Question

The sides \(\mathrm{AB}, \mathrm{BC}, \mathrm{CA}\) of a triangle \(\mathrm{ABC}\) have 3,4 and 5 interior points respectively on them the total no. of triangle that can be constructed by using these points as vertices is (a) 220 (b) 204 (c) 205 (d) 195

Short answer

The total number of triangles that can be constructed by using these points as vertices is 132.

Step-by-step solution

Case 1: All vertices on the same side

Since it is impossible for all three vertices to lie along the same side, there are no triangles in this case.

Case 2: Two vertices on one side and one vertex on another side

There are 3 combinations for this case: 1. Two vertices on AB and one vertex on BC: \(\binom{3}{2}\times\binom{4}{1} = 3\times4 = 12\) 2. Two vertices on BC and one vertex on CA: \(\binom{4}{2}\times\binom{5}{1} = 6\times5 = 30\) 3. Two vertices on CA and one vertex on AB: \(\binom{5}{2}\times\binom{3}{1} = 10\times3 = 30\) Total number of triangles for Case 2: \(12 + 30 + 30 = 72\)

Case 3: One vertex on each side

There are \(\binom{3}{1}\times\binom{4}{1}\times\binom{5}{1}\) combinations for this case: 1. One vertex on AB, one vertex on BC, and one vertex on CA: \(3\times4\times5 = 60\) Total number of triangles for Case 3: 60

Final answer

Sum up the number of triangles from all cases: Total triangles = Case 1 + Case 2 + Case 3 = 0 + 72 + 60 = 132 So, the given options do not have the correct answer. The correct answer should be 132 triangles can be constructed.

Key concepts

Triangle Basics

Triangles are a fundamental shape in geometry formed by three sides and three angles. In any given triangle, like the one in the exercise, the three points where the sides meet are called vertices. For a triangle to exist, these vertices cannot all be on the same line. This provides the shape and structure that allows for creating various triangles within the problem's context.

• A triangle has three sides, three vertices, and three angles.
• The sum of all interior angles in a triangle is always 180 degrees.
• In the problem, triangle ABC is formed with additional interior points on its sides.

Understanding these basic properties of triangles establishes a foundation for more complex topics, like combinatorics applied to geometry problems.

Understanding Binomial Coefficients

Binomial coefficients are a crucial tool in combinatorics, especially when determining the number of ways to select items from a group. These coefficients are often represented as \(\binom{n}{k}\), meaning the number of ways to choose \(k\) elements from a set of \(n\) elements.

• In our exercise, binomial coefficients are used to calculate combinations of points.
• Formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• They help understand how many triangles can be formed from the given points on each side.

With binomial coefficients, solving problems involving selections, such as those with points on a triangle's sides, becomes much more manageable.

Geometry and Triangle Construction

Geometry is the branch of mathematics focusing on the study of shapes, sizes, and the properties of space. When working with triangles, particularly in combination with points on sides, geometry helps us understand the spatial relationships and conditions needed to form a triangle.

• Points on triangle sides are used as vertices for forming new triangles.
• It involves calculating and validating conditions for each possible combination of points.
• In the exercise, geometric principles help exclude impossible configurations, like all points on the same line.

Using geometric principles allows for the systematic exploration and calculation of triangles, ensuring that only valid shapes are considered.

Vertices and Combinatorial Approaches

In a triangle, vertices are the points where two sides meet. For the problem presented, these vertices determine the different triangles that can be formed. With points placed on triangle sides, each set of three points can be selected as a potential set of vertices.

• Choosing different vertices changes the triangle's shape and size.
• Combinatorial approaches are used to count all possible configurations.
• In the provided solution, vertices from different sides are crucial for forming a valid triangle.

By exploring each potential set of vertices, we leverage combinatorial methods to accurately compute the total number of triangles that adhere to geometrical rules.