The distance formula is derived from the Pythagorean theorem.
The correct answer and explanation is:
Correct Answer: True
The distance formula is indeed derived from the Pythagorean Theorem. The Pythagorean Theorem is a fundamental principle in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It is expressed as:
a² + b² = c²
In coordinate geometry, the distance formula is used to determine the distance between two points in a plane. Suppose the two points are (x1,y1)(x₁, y₁) and (x2,y2)(x₂, y₂). To find the straight-line distance between them, imagine creating a right triangle by drawing a horizontal and a vertical line to connect the points. The horizontal side of the triangle is the difference in the x-values: x2−x1x₂ – x₁, and the vertical side is the difference in the y-values: y2−y1y₂ – y₁.
Applying the Pythagorean Theorem, the distance dd between the two points becomes:
d² = (x₂ – x₁)² + (y₂ – y₁)²
Taking the square root of both sides gives the distance formula:
d = √[(x₂ – x₁)² + (y₂ – y₁)²]
This formula is an application of the Pythagorean Theorem in the context of coordinate geometry. It works in two-dimensional space and can be extended to three-dimensional space by including the z-coordinates:
d = √[(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²]
Therefore, the distance formula is not only related to the Pythagorean Theorem but is a direct extension of it. It allows the calculation of the exact straight-line distance between any two points in a coordinate plane or space.