Pythagorean Theorem and distance between two points

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It can be expressed as: c² = a² + b².

A Pythagorean Triple consists of three positive integers (a, b, c) that satisfy the Pythagorean Theorem, meaning a² + b² = c². Example: (3, 4, 5) is a Pythagorean Triple.

The distance d can be found using the distance formula: d = √((x2 - x1)² + (y2 - y1)²).

Using the distance formula: d = √((4 - 1)² + (6 - 2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

Using the Pythagorean Theorem: c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, as stated by the Pythagorean Theorem.

Yes, because 5² + 12² = 25 + 144 = 169, and 13² = 169.

The area A of a square is calculated as A = side². So, A = 4² = 16.

The area A of a right triangle is given by A = (1/2) × base × height.

Using the Pythagorean Theorem: c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15.