Distance Formula — SAT Math Explained
A formula derived from the Pythagorean theorem that calculates the straight-line distance between two points in the coordinate plane: d = √[(x₂ - x₁)² + (y₂ - y₁)²].
The Core Idea
The distance between two points is the length of the hypotenuse of an invisible right triangle, with horizontal and vertical changes as the two legs. The Pythagorean theorem gives us this distance directly.
Step-by-Step: How to Approach Distance Formula
Identify coordinates of both points
Compute (x₂ - x₁)² — the horizontal component squared
Compute (y₂ - y₁)² — the vertical component squared
Add the two squared values
Take the square root
Simplify — express as exact radical or decimal approximation
Derivation
Between points (x₁, y₁) and (x₂, y₂):
Horizontal distance (run) = |x₂ - x₁|
Vertical distance (rise) = |y₂ - y₁|
These form legs of a right triangle
Hypotenuse (distance) = √(run² + rise²) = √[(x₂-x₁)² + (y₂-y₁)²]
Applications
Checking if a shape has equal sides (e.g., is a quadrilateral a rhombus?)
Verifying the Pythagorean theorem for specific triangles on a graph
Navigation and map problems
Common Errors to Avoid
Not squaring the differences before adding (can't just add and square at the end)
Getting a negative under the radical — check arithmetic (squares are always non-negative)
Confusing distance formula with midpoint formula
Practice: Distance Formula
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Distance from (0,0) to (3,4)?
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