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Calculate the distance between two points

Distance between two points in the Cartesian plane is the length of the line segment joining them, found from the horizontal and vertical changes using the Pythagorean theorem: d=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. The coordinates may produce exact radical or decimal lengths, and reversing the points leaves the distance unchanged; this treatment is limited to Euclidean distance in two dimensions, not three-dimensional or non-Euclidean settings.

Detailed Explanation: Calculate the distance between two points

The distance between two points is the length of the line segment connecting them. For points (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2), use the distance formula:

d=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

This formula uses the horizontal change, x2−x1x_2-x_1, and the vertical change, y2−y1y_2-y_1.

Example: Find the distance between A(−2,1)A(-2,1) and B(4,9)B(4,9).

  1. Identify the coordinates:

(x1,y1)=(−2,1),(x2,y2)=(4,9) (x_1,y_1)=(-2,1), \qquad (x_2,y_2)=(4,9)

  1. Substitute into the formula:

d=(4−(−2))2+(9−1)2 d=\sqrt{(4-(-2))^2+(9-1)^2}

  1. Simplify the differences:

d=62+82 d=\sqrt{6^2+8^2}

  1. Square and add:

d=36+64=100 d=\sqrt{36+64}=\sqrt{100}

  1. Simplify:

d=10 d=10

The distance between AA and BB is 10\boxed{10} units. Reversing the order of the points gives the same distance because the differences are squared.

Learn by doing: Calculate the distance between two points

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    Cartesian Grid - Distance as Radical Between Coordinates (Angle)

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