Pythagorean Theorem

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.

The legs of a right triangle are the two sides that form the right angle. They are usually referred to as 'a' and 'b' in the Pythagorean Theorem.

Using the Pythagorean Theorem: c² = 5² + 12² = 25 + 144 = 169. Therefore, c = √169 = 13 cm.

Using the Pythagorean Theorem: 10² = 6² + b². Thus, 100 = 36 + b², so b² = 64, and b = √64 = 8 cm.

Using the Pythagorean Theorem: 15² = 9² + b². Thus, 225 = 81 + b², so b² = 144, and b = √144 = 12 cm.

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2. The shortest side is opposite the 30° angle, the longer leg is opposite the 60° angle, and the hypotenuse is the longest side.

In a 45-45-90 triangle, the lengths of the legs are equal, and the hypotenuse is √2 times the length of each leg.

Using the Pythagorean Theorem: c² = 8² + 15² = 64 + 225 = 289. Therefore, c = √289 = 17 cm.

The distance formula between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane is: d = √((x₂ - x₁)² + (y₂ - y₁)²).

Using the Pythagorean Theorem: 25² = 20² + b². Thus, 625 = 400 + b², so b² = 225, and b = √225 = 15 cm.