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: a² + b² = c².

The hypotenuse is the longest side of a right triangle, opposite the right angle.

To find a missing side, rearrange the Pythagorean Theorem: a² = c² - b² or b² = c² - a², depending on which side is missing.

Using the Pythagorean Theorem: b² = c² - a²; b² = 20² - 14²; b² = 400 - 196; b² = 204; b = √204 ≈ 14.4.

The approximate value of the square root of 20 is about 4.5.

To estimate the square root, find two perfect squares between which the number lies and then approximate based on their square roots.

In a 3-4-5 triangle, the sides are in the ratio of 3:4:5, which satisfies the Pythagorean Theorem (3² + 4² = 5²).

The Pythagorean Theorem is used in various fields such as construction, navigation, and physics to calculate distances and angles.

In coordinate geometry, the distance between two points (x₁, y₁) and (x₂, y₂) can be found using the formula: d = √((x₂ - x₁)² + (y₂ - y₁)²).

Using the Pythagorean Theorem: c = √(6² + 8²) = √(36 + 64) = √100 = 10.