CBSE Class 7 Maths: The Triangles and its Properties Ex 6.4

Welcome, young mathematicians! In this important lesson, we're diving deep into "The Triangles and its Properties," specifically focusing on Exercise 6.4. Triangles are fundamental shapes in geometry, and understanding their special rules is key to solving many real-world problems.

This exercise introduces a fascinating property called the Triangle Inequality Theorem. It's not just a fancy name; it's a rule that helps us determine if a set of three lengths can actually form a triangle. Imagine trying to build a triangle with three sticks – sometimes they just don't connect! By the end of this page, you'll not only understand this crucial property but also be able to confidently apply it to check if a triangle is possible and solve related problems, making you a geometry expert!

What is a Triangle? A Quick Recap

Before we explore advanced properties, let's quickly refresh our memory on what a triangle is. A triangle is a closed, two-dimensional shape with three straight sides and three angles. It's the simplest polygon! The points where the sides meet are called vertices. For any triangle, the sum of its three interior angles is always 180 degrees. Understanding these basic elements is crucial for grasping how the sides relate to each other, especially when we talk about whether a triangle can even exist with certain side lengths. Think of a triangle as a basic building block in geometry; its properties are fundamental to more complex shapes and structures around us, from bridges to roof trusses.

The Triangle Inequality Theorem: The Golden Rule

This is the core concept of Exercise 6.4 and one of the most important properties of triangles! The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's break this down:

Imagine you have three sticks of lengths 'a', 'b', and 'c'. To form a triangle, three conditions must be met:

  1. a + b > c
  2. b + c > a
  3. a + c > b

If even one of these conditions is not true, you cannot form a triangle with those side lengths. Think about it: if two sides are too short, they won't reach each other to connect and form the third vertex. For example, if you have sticks of length 2 cm, 3 cm, and 6 cm. If you try to join the 2 cm and 3 cm sticks, their combined length (5 cm) is less than 6 cm, so they can't possibly meet at the top to form the third corner! This theorem is vital because it sets the very condition for a triangle's existence, making it a foundational principle in geometry and its applications.

How to Check if Side Lengths Can Form a Triangle

  1. Identify the side lengths — Let the given three side lengths be 'a', 'b', and 'c'. For example, if you are given 5 cm, 7 cm, and 10 cm.
  2. Check the first pair — Add the lengths of the first two sides and compare it with the third side. Is a + b > c? For our example: 5 cm + 7 cm = 12 cm. Is 12 cm > 10 cm? Yes, it is.
  3. Check the second pair — Now, add the lengths of the second and third sides and compare it with the first side. Is b + c > a? For our example: 7 cm + 10 cm = 17 cm. Is 17 cm > 5 cm? Yes, it is.
  4. Check the third pair — Finally, add the lengths of the first and third sides and compare it with the second side. Is a + c > b? For our example: 5 cm + 10 cm = 15 cm. Is 15 cm > 7 cm? Yes, it is.
  5. Conclude based on all checks — If ALL THREE conditions are true, then a triangle CAN be formed with the given side lengths. If even one condition is false, then a triangle CANNOT be formed. In our example, since all three conditions were true, a triangle can be formed with sides 5 cm, 7 cm, and 10 cm.

Exam Tip: Don't Forget All Three Checks!

A very common mistake students make is checking only one or two of the three possible sums. Remember, the Triangle Inequality Theorem requires that **the sum of any two sides** must be greater than the third side. It's not enough if a + b > c is true; you must also verify b + c > a and a + c > b. If even one of these three conditions fails, then the triangle cannot be formed. Always write down all three inequalities and test them systematically to avoid losing marks. This thoroughness ensures you apply the theorem correctly every time, especially in tricky questions where only one condition might fail.

Practice Questions with Solutions

  • Q: Is it possible to have a triangle with the following sides: 3 cm, 4 cm, 5 cm? A: Step 1: Check the first pair: 3 + 4 = 7. Is 7 > 5? Yes. Step 2: Check the second pair: 4 + 5 = 9. Is 9 > 3? Yes. Step 3: Check the third pair: 3 + 5 = 8. Is 8 > 4? Yes. Final answer: Since all three conditions are satisfied, it is possible to have a triangle with sides 3 cm, 4 cm, 5 cm.
  • Q: Can a triangle be formed with side lengths 2 cm, 5 cm, 8 cm? A: Step 1: Check the first pair: 2 + 5 = 7. Is 7 > 8? No. Step 2: Since one condition fails, there is no need to check further. A triangle cannot be formed. Final answer: It is not possible to have a triangle with sides 2 cm, 5 cm, 8 cm.
  • Q: Two sides of a triangle are 6 cm and 8 cm. Between which two numbers can the length of the third side fall? A: Step 1: Let the third side be 'x'. According to the triangle inequality: Condition 1: 6 + 8 > x => 14 > x Condition 2: 6 + x > 8 => x > 8 - 6 => x > 2 Condition 3: 8 + x > 6 => x > 6 - 8 => x > -2 (This is always true as length cannot be negative, so we focus on x > 2). Step 2: Combine the valid inequalities from Step 1. Final answer: The length of the third side (x) must be greater than 2 cm and less than 14 cm (i.e., 2 < x < 14).
  • Q: Determine if a triangle can be formed with sides of lengths 10 cm, 10 cm, and 10 cm. A: Step 1: Check the first pair: 10 + 10 = 20. Is 20 > 10? Yes. Step 2: Check the second pair: 10 + 10 = 20. Is 20 > 10? Yes. Step 3: Check the third pair: 10 + 10 = 20. Is 20 > 10? Yes. Final answer: Yes, a triangle (specifically an equilateral triangle) can be formed with sides of 10 cm, 10 cm, and 10 cm.

Frequently Asked Questions

What is the main property discussed in 'The Triangles and its Properties Ex 6.4'?

The main property is the Triangle Inequality Theorem. It states that the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. This theorem is fundamental for determining if a set of three lengths can actually form a triangle.

Why is the Triangle Inequality Theorem important?

It is important because it defines the very condition for the existence of a triangle. Without this property, three line segments cannot connect to form a closed three-sided figure. It has practical applications in geometry, construction, and pathfinding, ensuring shapes are structurally sound or paths are efficient.

What if the sum of two sides is exactly equal to the third side?

If the sum of two sides is exactly equal to the third side, then a triangle cannot be formed. The three points would lie on a straight line, forming a degenerate triangle, not a proper triangle with distinct vertices. The condition must be strictly 'greater than', not 'greater than or equal to'.

How many checks do I need to perform to verify the Triangle Inequality Theorem?

You need to perform three checks, one for each possible pair of sides. You must ensure that (side1 + side2 > side3), (side2 + side3 > side1), and (side1 + side3 > side2). If all three conditions are met, then a triangle can be formed.