Which expression correctly gives the distance between two points?

Which expression correctly gives the distance between two points?

• A. d = |x2 - x1| + |y2 - y1|
• B. d = sqrt[(x2 + x1)^2 + (y2 - y1)^2]
• C. d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
• D. d = sqrt[(x2 - x1)^2 - (y2 - y1)^2]

Answer: (C)

The distance between two points in the plane is found by using the horizontal and vertical differences between the points. If you move from (x1, y1) to (x2, y2), the horizontal change is Δx = x2 − x1 and the vertical change is Δy = y2 − y1. These two changes form the legs of a right triangle, and the distance is the length of the hypotenuse. By the Pythagorean theorem, d^2 = (Δx)^2 + (Δy)^2, so the distance is d = sqrt[(x2 − x1)^2 + (y2 − y1)^2]. This expression is the correct way to measure straight-line distance. Other forms don’t match this geometric setup: using the sum of absolute differences gives the grid-based (Manhattan) distance, not the direct line; using a sum inside the square (x2 + x1)^2 mixes coordinates and loses the essential difference; and taking the square root of a difference of squares (a^2 − b^2) isn’t how the Euclidean distance is computed and can be invalid.

The distance between two points in the plane is found by using the horizontal and vertical differences between the points. If you move from (x1, y1) to (x2, y2), the horizontal change is Δx = x2 − x1 and the vertical change is Δy = y2 − y1. These two changes form the legs of a right triangle, and the distance is the length of the hypotenuse. By the Pythagorean theorem, d^2 = (Δx)^2 + (Δy)^2, so the distance is d = sqrt[(x2 − x1)^2 + (y2 − y1)^2].

This expression is the correct way to measure straight-line distance. Other forms don’t match this geometric setup: using the sum of absolute differences gives the grid-based (Manhattan) distance, not the direct line; using a sum inside the square (x2 + x1)^2 mixes coordinates and loses the essential difference; and taking the square root of a difference of squares (a^2 − b^2) isn’t how the Euclidean distance is computed and can be invalid.