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Determine 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 coordinate differences using the Pythagorean relationship: d=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. The understanding includes handling negative coordinates, recognizing that reversing the points does not change the distance, and expressing results exactly or approximately, with horizontal and vertical cases as special instances; it is confined to Euclidean two-dimensional coordinates, not three-dimensional or more abstract distance measures.

Detailed Explanation: Determine the distance between two points

The distance between two points is the length of the line segment joining them. Use the distance formula:

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

This formula comes from the Pythagorean theorem: the horizontal difference and vertical difference form the two legs of a right triangle.

Example: Find the distance between A(−2,3)A(-2,3) and B(4,−4)B(4,-4).

  1. Identify the coordinates:

(x1,y1)=(−2,3),(x2,y2)=(4,−4) (x_1,y_1)=(-2,3), \qquad (x_2,y_2)=(4,-4)

  1. Find the horizontal and vertical differences:

x2−x1=4−(−2)=6 x_2-x_1=4-(-2)=6 y2−y1=−4−3=−7 y_2-y_1=-4-3=-7

  1. Substitute into the distance formula:

d=62+(−7)2 d=\sqrt{6^2+(-7)^2}

  1. Simplify:

d=36+49=85 d=\sqrt{36+49}=\sqrt{85}

Therefore, the exact distance is

85 units\boxed{\sqrt{85}\text{ units}}

As a decimal, this is approximately

9.22 units\boxed{9.22\text{ units}}

The order of the points does not matter. Reversing them changes the signs of the differences, but squaring makes the final distance the same.

Learn by doing: Determine the distance between two points

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    Cartesian Grid - Distance as Radical Between Coordinates (Angle)

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