Mathematics is full of important theorems that help students solve problems easily. If you are a student of Class 10-12, understanding these theorems can make math simpler and more interesting. Here are 10 must-know theorems that will help you in exams and beyond!
1. Pythagoras Theorem
📌 Formula: a2+b2=c2a^2 + b^2 = c^2a2+b2=c2
✔ Used in: Right-angled triangles
👉 Example: If one side is 3 cm, the other is 4 cm, the hypotenuse is 5 cm (since 32+42=523^2 + 4^2 = 5^232+42=52).
2. Thales’ Theorem (Basic Proportionality Theorem)
📌 Statement: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
✔ Used in: Similar triangles, geometry proofs
3. Heron’s Formula
📌 Formula: A=s(s−a)(s−b)(s−c)A = \sqrt{s(s-a)(s-b)(s-c)}A=s(s−a)(s−b)(s−c)​
✔ Used in: Finding the area of a triangle when all sides are known
👉 Example: If sides are a = 5 cm, b = 6 cm, c = 7 cm, first find s = (5+6+7)/2 = 9 cm, then use the formula to get the area.
4. Midpoint Theorem
📌 Statement: The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
✔ Used in: Coordinate geometry, triangle properties
5. Area Theorem of a Triangle
📌 Formula: 12×base×height\frac{1}{2} \times \text{base} \times \text{height}21​×base×height
✔ Used in: Finding the area of triangles easily
6. Quadratic Equation Theorem
📌 Formula: ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0
📌 Solution: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​
✔ Used in: Solving quadratic equations in algebra
7. Sum of Angles in a Triangle
📌 Statement: The sum of the interior angles of a triangle is always 180°.
✔ Used in: Geometry, proving other theorems
8. Circle Theorems (Tangent and Secant Theorem)
📌 Statement: The tangent to a circle is always perpendicular to the radius at the point of contact.
✔ Used in: Circle geometry, problem-solving
9. Binomial Theorem
📌 Formula: (a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k(a+b)n=∑k=0n​(kn​)an−kbk
✔ Used in: Expanding algebraic expressions, higher math concepts
10. Lagrange’s Mean Value Theorem
📌 Statement: If a function is continuous and differentiable, then there exists a point where the tangent is parallel to the secant line.
✔ Used in: Calculus, motion problems
Conclusion
These theorems are fundamental in mathematics and are frequently used in exams. Make sure to practice them with examples to understand better. Which one do you find the most interesting? Let us know in the comments! 📢💡
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