Geometric Proofs and Proof by Exhaustion

This page delves into geometric proofs and introduces the concept of proof by exhaustion.

A geometric proof example is provided to demonstrate that A(1,1), B(3,3), and C(4,2) form a right-angled triangle. The proof involves calculating gradients of lines and showing that two lines are perpendicular.

Example: To prove ABC is a right-angled triangle, the gradients of AB and BC are calculated and shown to be perpendicular (product of gradients = -1).

Proof by exhaustion is then introduced as a method where all possible cases are considered and proven individually.

Definition: Proof by exhaustion involves splitting a problem into smaller cases and tackling each one separately.

An example of proof by exhaustion is given to show that all square numbers are either a multiple of 4 or one more than a multiple of 4.

Highlight: When the set of values being considered is small, you can try all cases numerically in a proof by exhaustion.

Proof by Counter-Example and Completing the Square

This page covers proof by counter-example and introduces the use of completing the square in proofs.

Proof by counter-example involves finding one instance where a statement does not hold true to disprove it.

Example: To disprove "the sum of two consecutive prime numbers is always even," the counter-example of 2 + 3 = 5 (an odd number) is provided.

The page then demonstrates how to use completing the square to prove that an expression is always positive.

Example: Using completing the square to prove that n² - 8n + 20 is always positive for all values of n.

Highlight: How to use completing the square for proof is demonstrated as an effective technique for showing an expression's behavior across all values.