List of Important Theorems in Maths with Statements and Uses
The concept of theorems in mathematics is essential for solving problems, understanding proofs, and excelling in board exams like CBSE, ICSE, and various competitive tests. A good grasp on key maths theorems supports logical reasoning and builds confidence in approaching questions about geometry, algebra, and calculus.
What Is Theorem in Mathematics?
A theorem in mathematics is a statement that can be logically proven to be true using previously established results, like axioms and definitions. Theorems are crucial in topics such as geometry, algebra, and calculus, forming the backbone of mathematical knowledge. While axioms and postulates are assumed to be true, theorems require proof and connect ideas across various chapters.
Types of Theorems in Maths
There are many types of theorems in mathematics. They are often grouped by subject or the shapes they discuss. Below is a quick table of the main types of theorems students commonly encounter:
| Type | Examples | Board Relevance |
|---|---|---|
| Geometry Theorems | Pythagoras, Circles, Similar Triangles, Midpoint | Class 9, 10 |
| Algebra Theorems | Remainder, Factor, Fundamental Theorem of Arithmetic | Class 9, 10 |
| Calculus Theorems | Mean Value, Rolle’s, Lagrange | Class 11, 12 |
| Probability Theorems | Bayes, Total Probability | Class 10, 12 |
Key Formulae and Statements
Here are some of the most important theorems in maths for class 10 and above:
| Theorem Name | Statement / Formula | Use |
|---|---|---|
| Pythagoras Theorem | \( c^2 = a^2 + b^2 \) | Right-angled triangles |
| Midpoint Theorem | Line joining midpoints is parallel/half the third side | Triangles |
| Remainder Theorem | Remainder = \( f(a) \) when dividing by \( x-a \) | Polynomials |
| Angle Bisector Theorem | Angle bisector divides side in ratio of other sides | Triangles |
| Circle Theorems | Angles, chords, tangents rules | Circles |
Step-by-Step Illustration: Example—Pythagoras Theorem
Let’s apply the Pythagoras theorem to solve a simple problem:
Given: A right triangle has sides 6 cm and 8 cm. Find the hypotenuse.
1. Identify sides: Let \( a = 6 \) cm, \( b = 8 \) cm.2. Use formula: \( c^2 = a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100 \).
3. Take square root: \( c = \sqrt{100} = 10 \) cm.
4. Final Answer: Hypotenuse = 10 cm.
Common Mistakes To Avoid
- Mixing up theorem and axiom—remember, theorems need proof!
- Forgeting required geometric conditions (e.g., using Pythagoras in a non-right triangle).
- Missing steps in proofs, or incorrect diagram labeling.
- Assuming all circles follow the same theorem for angles without checking if the center or circumference is used.
Relation To Other Concepts
The concept of theorems in mathematics is directly related to topics like postulates (statements accepted without proof) and triangle properties. Knowing which theorem to apply is key to solving geometry questions and understanding advanced proofs, as seen in competitive exams and higher grades.
Quick Revision Table for Circle Theorems
| Circle Theorem | Statement |
|---|---|
| Angle at Centre Theorem | Angle at centre is twice angle at circumference |
| Equal Chords | Equal chords subtend equal angles at the centre |
| Perpendicular to a Chord | Perpendicular from centre to chord bisects the chord |
| Cyclic Quadrilateral | Opposite angles sum to 180° |
Try These Yourself!
- List three geometry theorems you remember and state their formulas.
- Try proving the midpoint theorem using a diagram.
- Identify a theorem needed to show that two triangles are similar.
- Check which circle theorem applies if two chords are the same length.
Speed Tricks for Theorems in Mathematics
Many students find it quick to remember regular theorem forms as “if...then...” statements. For example: If a triangle is right-angled, then Pythagoras theorem applies. Visual memory cues (highlighting corresponding sides) also minimize errors. Vedantu’s teachers use stepwise sketches and color-coding to teach quick recognition during live online sessions.
Wrapping It All Up
We have learned how theorems in mathematics are powerful tools for solving problems, especially in geometry and circle questions. Practice writing proofs and connecting conditions for fast revision. Vedantu makes mastering theorems easy so you can approach every exam confidently!
FAQs on Theorems in Mathematics: List, Proofs & Examples
1. What are the important theorems in Mathematics Class 10?
Class 10 mathematics covers several crucial theorems. Key examples include the Pythagoras Theorem, the Midpoint Theorem, the Remainder Theorem, the Fundamental Theorem of Arithmetic, the Angle Bisector Theorem, and theorems related to circles (such as the inscribed angle theorem). These theorems are fundamental to understanding geometry, algebra, and number systems, and are frequently tested in examinations.
2. What are the important theorems of triangles in Class 10?
Understanding triangle theorems is essential for Class 10 math. Key theorems include those related to similarity (if corresponding angles are equal, and corresponding sides are in the same ratio, then triangles are similar), congruence (congruent triangles are also similar, but not vice-versa), and the application of these concepts in problem-solving. Mastering these concepts is crucial for geometry-based questions.
3. What are the important circle theorems in Class 10?
Circle theorems form a significant part of Class 10 geometry. Important theorems cover relationships between chords, angles subtended by arcs, and the properties of cyclic quadrilaterals. For example, remember that equal chords subtend equal angles at the center, and the perpendicular from the center to a chord bisects the chord. Understanding these relationships is vital for solving problems involving circles.
4. What should I know about theorems of triangles in Class 10?
Theorems related to triangles are foundational to geometry. A strong grasp of theorems concerning similarity and congruence is vital for solving various geometry problems. Remember to understand how these theorems are applied in different contexts within the syllabus and practice applying them to different problem types. This is a high-weightage topic for the Class 10 exams.
5. Where can I find simplified explanations of Class 10 triangle theorems?
Vedantu provides comprehensive and simplified explanations of Class 10 triangle theorems. Our resources include detailed explanations, solved examples, and practice exercises designed to help students understand and master these concepts effectively. Our expert-created materials are designed to help you succeed in your exams.
6. How are theorems different from postulates and axioms?
While all three are fundamental statements in mathematics, they differ in their nature. Axioms are self-evident truths that form the foundation of a mathematical system. Postulates are assumptions specific to a particular geometry (like Euclidean geometry). Theorems, on the other hand, are statements that are proven true using logic and previously established axioms, postulates, or other theorems.
7. What is the Pythagorean Theorem and how is it applied?
The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is expressed as a² + b² = c², where 'c' is the hypotenuse. It's widely applied in solving problems involving right-angled triangles, finding distances, and in various areas of geometry and trigonometry.
8. Explain the Midpoint Theorem.
The Midpoint Theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length. This theorem is useful in proving other geometric relationships and solving problems involving parallel lines and triangles.
9. What is the Remainder Theorem and how does it work?
The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is f(a). This means you can find the remainder of a polynomial division without actually performing the long division. Simply substitute 'a' into the polynomial to find the remainder.
10. What is the Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic (also known as the Unique Factorization Theorem) states that every integer greater than 1 can be uniquely represented as a product of prime numbers. This is a cornerstone of number theory and has significant implications in various mathematical fields.
11. How can I improve my understanding of theorems?
To improve your understanding of theorems, focus on: *Understanding the definitions and statements clearly; *Following the proofs step by step; *Practicing solving problems that apply the theorems; *Visualising the theorems using diagrams and real-world examples; and *Seeking help from teachers or online resources when needed.
12. Are there practice resources available for Class 10 theorems?
Yes, Vedantu offers a wealth of practice resources for Class 10 theorems. Our resources include solved examples, practice problems, and mock tests designed to reinforce your understanding and improve your problem-solving skills. These resources will help prepare you for your exams.
