The Pythagorean Theorem and Coordinate Distance

Objective: Apply the Pythagorean theorem to perpendicular legs.

Prerequisite skills: squares, square roots, and coordinate pairs. In a right triangle with legs a,b and hypotenuse c, a²+b²=c². The hypotenuse is the longest side and lies opposite the 90° angle. Take the nonnegative root for a physical length. Common error: adding the leg lengths rather than their squares. Restrictions and interpretation: The hypotenuse is opposite the right angle; the Pythagorean equation does not apply to an arbitrary nonright triangle.

Objective: Subtract a known leg square from the hypotenuse square.

When the hypotenuse and one leg are known, rearrange a²+b²=c² to isolate the missing square. A result less than zero indicates inconsistent dimensions for a right triangle. Common error: adding squares when the given longest side is already the hypotenuse. Restrictions and interpretation: The hypotenuse is opposite the right angle; the Pythagorean equation does not apply to an arbitrary nonright triangle.

Objective: Use horizontal and vertical coordinate differences as right-triangle legs.

The segment between two points is the hypotenuse of a right triangle whose legs are the coordinate differences. Squaring removes the signs, so consistent coordinate subtraction yields a nonnegative distance. Common error: using coordinate sums instead of differences. Restrictions and interpretation: The hypotenuse is opposite the right angle; the Pythagorean equation does not apply to an arbitrary nonright triangle. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.