Overview
This lecture covers Pythagoras’ theorem, its application to right-angled triangles, methods to prove the theorem, how to identify triangle sides, Pythagorean triplets, and real-life and exam-style problems using the theorem.
Introduction to Pythagoras' Theorem
- Pythagoras’ theorem applies to right-angled triangles, where one angle is 90 degrees.
- The side opposite the 90-degree angle is the hypotenuse; the other two are called legs.
- The theorem states: in a right-angled triangle, the area (square) on the hypotenuse equals the sum of the areas (squares) on the other two sides, or ( a^2 + b^2 = c^2 ).
Identifying Sides in Right Triangles
- The hypotenuse is always opposite the right angle and is the triangle's longest side.
- The other two sides (legs) are adjacent to the right angle.
- To find the hypotenuse or a missing leg, use the formula rearranged as needed.
Basic Algebraic Tools for Pythagoras
- When squaring binomials: ( (a+b)^2 = a^2 + 2ab + b^2 ) and ( (a-b)^2 = a^2 - 2ab + b^2 ).
- To square fractions: ( \left( \frac{a}{b} \right)^2 = \frac{a^2}{b^2} ).
- To find area of a right triangle: ( \frac{1}{2} \times \text{base} \times \text{height} ).
Pythagoras’ Theorem Explained and Proven
- In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
- The relationship can be proven geometrically by comparing areas or using similar triangles.
Applications & Solving Problems
- To find the hypotenuse: ( c = \sqrt{a^2 + b^2} ).
- To find a leg: ( a = \sqrt{c^2 - b^2} ).
- Real-life examples: carpenters, masons, and others use the theorem for constructing right angles and calculating lengths.
Pythagorean Triplets
- Sets like (3, 4, 5), (5, 12, 13), and (7, 24, 25) are Pythagorean triplets: they satisfy ( a^2 + b^2 = c^2 ).
- Triplets can be generated by multiplying known triplets by any positive integer.
- Patterns exist for constructing triplets using ( n^2 - 1,\ n, n^2 + 1 ) and similar formulas.
Additional Theorems and Extensions
- The converse of Pythagoras: If ( a^2 + b^2 = c^2 ), then the triangle is right-angled.
- Related theorems apply to squares, rectangles, rhombuses, and equilateral triangles when analyzing sides and diagonals.
Types of Exam Questions
- Identify missing sides using the theorem.
- Prove properties in geometric shapes using Pythagoras.
- Calculate lengths in real-life or application-based settings.
- Construct Pythagorean triples or use tables to verify right-angled triangles.
Key Terms & Definitions
- Pythagoras' Theorem — In a right-angled triangle, ( a^2 + b^2 = c^2 ), where c is the hypotenuse.
- Hypotenuse — The side opposite the right angle in a right-angled triangle.
- Leg (of triangle) — Either of the two sides adjacent to the right angle.
- Pythagorean Triplet — A set of three positive integers that satisfy ( a^2 + b^2 = c^2 ).
Action Items / Next Steps
- Complete exercises 17.1, 17.2, and 17.3 as assigned in the lecture.
- Practice identifying hypotenuse and legs in various triangles.
- Memorize common Pythagorean triplets (e.g., 3-4-5, 5-12-13).
- Apply Pythagoras’ theorem to solve real-life and geometric problems.