Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Triangles ↓


Inequalities in a Triangle


Triangles are three-sided polygons that are an essential part of geometry. They come in different shapes and sizes, but they all follow some basic rules. One important set of rules involves the "inequality in triangles." These inequalities are rules or properties that relate the lengths of the sides of a triangle to each other. Understanding these inequalities helps us solve problems involving triangles and assess whether the given lengths can form a triangle.

Basic triangle structure

Before we discuss inequalities, let's review the basic properties of a triangle. Every triangle has three sides, three angles, and the sum of the angles is 180 degrees. Triangles can be classified into different types based on their sides and angles:

  • Equilateral triangle: All sides are equal, and all angles are 60 degrees.
  • Isosceles triangle: Two sides are equal, and the angles opposite to these sides are also equal.
  • Scalene triangle: All sides and angles are different.

Triangle inequality theorem

The most basic inequality involving triangles is the Triangle Inequality Theorem. This 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. Mathematically, if we have a triangle with sides a, b, and c, the theorem can be expressed in three inequalities:

a + b > c
a + c > b
b + c > a

This theorem ensures that the sides form a closed figure rather than a straight line.

Visual example

To understand these inequalities, consider the following examples of triangles.

C A B

In the above triangle:

  • Let side a be 6 units, b is 8 units, and c is 10 units.
  • a + b = 6 + 8 = 14 units, which is more than c (10 units).
  • b + c = 8 + 10 = 18 units, which is more than a (6 units).
  • c + a = 10 + 6 = 16 units, which is more than b (8 units).

Since all the conditions satisfy the inequality, these sides can form a triangle.

Example problems

Let's solve an example problem:

Example 1: Determine if sides of length 4, 6, and 9 can form a triangle.

Using the Triangle Inequality Theorem, we can check three possible inequalities:

4 + 6 > 9

Its simplification is 10 > 9, which is true.

4 + 9 > 6

Its simplification is 13 > 6, which is also true.

6 + 9 > 4

Its simplification is 15 > 4, which is also true.

Since all three conditions are true, sides of length 4, 6, and 9 can indeed form a triangle.

Example 2: Can the lengths 2, 2, and 5 together form a triangle?

Let us verify using inequalities:

2 + 2 > 5

Its simplification is 4 > 5, which is false.

Since one of these conditions is not met, these lengths cannot form a triangle.

Understanding the concept with more complex examples

When dealing with triangles, especially in real-world contexts or more advanced problems, it becomes important to understand these inequalities in depth. Consider more complex cases where an understanding of triangle inequalities can aid in problem-solving.

Example 3: If a = 4, b = 7, and c is unknown, what is the range of possible values for c to form a valid triangle?

For c to form a valid triangle with a and b, it must satisfy the following:

4 + 7 > c

It's simple: c < 11.

4 + c > 7

It's simple: c > 3.

7 + c > 4

This condition, c > -3, is always satisfied if c > 3.

Thus, c must be greater than 3 and less than 11. The possible side lengths for c must lie within the interval (3, 11).

Key points to remember

Here are some important points to remember about inequalities in triangles:

  • The triangle inequality theorem ensures that vertices can form a triangle, but not be collinear points.
  • Understanding these inequalities can help determine the possible range of the unknown side of a triangle.
  • Representing these inequalities diagrammatically can simplify complex problems.

Conclusion

Inequalities in triangles play an important role in understanding the structure and properties of triangles in mathematics. They help us determine the feasibility of constructing a triangle with given sides and provide a basis for solving more complex geometric problems. Mastering these concepts opens the way to deeper geometric studies, such as those involving trigonometry and calculus.

Keep practicing the application of triangle inequalities with a variety of examples to build a stronger understanding and intuition for recognizing feasible triangles in different problem situations.


Grade 9 → 7.4


U
username
0%
completed in Grade 9

← Prev (7.3)
Properties of Triangles
Next (7.5) →
Similarity of Triangles

Comments 



Inequalities in a Triangle

https://www.buddymath.com/g9/7.4?bmal=en