Pythagorean Theorem

Formula (2D - Right Triangle)

Formula: a² + b² = c²

Explanation: 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).

Step-by-Step Guide (2D)

To solve for the sides of a right triangle using the Pythagorean Theorem:

Step 1: Identify the lengths of the two sides (a and b) that are not the hypotenuse. These will be given or calculated.
Step 2: Use the formula a² + b² = c² to find the hypotenuse (c), or rearrange the formula to find one of the legs (a or b).
Step 3: Take the square root of the result to get the length of the missing side.

Example: If side a is 3 and side b is 4, find the hypotenuse (c):

a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25 = 5

Formula (3D - Distance in Space)

Formula: d = √(x² + y² + z²)

Explanation: This formula is used to calculate the distance between two points in three-dimensional space, where x, y, and z represent the differences in the coordinates of the points in the x, y, and z directions.

Step-by-Step Guide (3D)

To solve for the distance between two points in 3D space:

Step 1: Identify the coordinates of the two points in space, represented as (x₁, y₁, z₁) and (x₂, y₂, z₂).
Step 2: Find the differences in the x, y, and z coordinates: Δx = x₂ - x₁, Δy = y₂ - y₁, and Δz = z₂ - z₁.
Step 3: Apply the 3D distance formula: d = √(Δx² + Δy² + Δz²) to find the distance.

Example: Find the distance between the points (1, 2, 3) and (4, 6, 8):

Δx = 4 - 1 = 3
Δy = 6 - 2 = 4
Δz = 8 - 3 = 5
d = √(3² + 4² + 5²) = √(9 + 16 + 25) = √50 ≈ 7.07

Additional Information

The Pythagorean Theorem works for both 2D and 3D right-angled triangles and can be used to calculate the distance between points in any dimensional space.

In 2D, the formula applies to right triangles, and in 3D, it’s useful for finding the distance between two points in space. You can use this for practical applications like physics and geometry problems.

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