# Distance between two points

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Definition.

**Distance between two points**is the length of the line segment that connects this points.## Distance formulas:

*The formula for calculating the distance between two points*A(x_{a}, y_{a}) and B(x_{b}, y_{b}) on plane:

AB = √(x _{b}- x_{a})^{2}+ (y_{b}- y_{a})^{2}*The formula for calculating the distance between two points*A(x_{a}, y_{a}, z_{a}) and B(x_{b}, y_{b}, z_{b}) in space:

AB = √(x _{b}- x_{a})^{2}+ (y_{b}- y_{a})^{2}+ (z_{b}- z_{a})^{2}

## Proof of the formula of distance between two points for the plane problem

From the points A and B drop perpendiculars to the coordinate axes.

Consider a right triangle ∆ABC. Legs of the triangle are equal to:

_{b}- x

_{a};

BC = y

_{b}- y

_{a}.

Using the Pythagorean theorem, calculate the length of the hypotenuse AB:

^{2}+ BC

^{2}.

Substituting to this expression the lengths of AC and BC which expressed in terms of the coordinates of points A and B, we obtain the formula for calculating the distance between points on the plane.

Proof of the formula for calculating the distance between two points in space is similar.

## Examples of tasks with distance between two points

### Examples of tasks with distance between two points on 2D

Example 1.

Find the distance between two points A(-1, 3) and B(6,2).
**Solution.**

_{b}- x

_{a})

^{2}+ (y

_{b}- y

_{a})

^{2}= √(6 - (-1))

^{2}+ (2 - 3)

^{2}= √7

^{2}+ 1

^{2}= √50 = 5√2

**Answer:** AB = 5√2.

Example 2.

Find the distance between two points A(0, 1) and B(2,-2).
**Solution.**

AB = √(x

_{b}- x_{a})^{2}+ (y_{b}- y_{a})^{2}= √(2 - 0)^{2}+ (-2 - 1)^{2}= √2^{2}+ (-3)^{2}= √13**Answer:** AB = √13.

### Examples of tasks with distance between two points on 3D

Example 3.

Find the distance between two points A(-1, 3, 3) and B(6, 2, -2).
**Solution.**

_{b}- x

_{a})

^{2}+ (y

_{b}- y

_{a})

^{2}+ (z

_{b}- z

_{a})

^{2}=

= √(6 - (-1))

^{2}+ (2 - 3)

^{2}+ (-2 - 3)

^{2}= √7

^{2}+ 1

^{2}+ 5

^{2}= √75 = 5√3

**Answer:** AB = 5√3.

Example 4.

Find the distance between two points A(0, -3, 3) and B(3, 1, 3).
**Solution.**

AB = √(x

= √(3 - 0)

_{b}- x_{a})^{2}+ (y_{b}- y_{a})^{2}+ (z_{b}- z_{a})^{2}== √(3 - 0)

^{2}+ (1 - (-3))^{2}+ (3 - 3)^{2}= √3^{2}+ 4^{2}+ 0^{2}= √25 = 5**Answer:** AB = 5.

Analytic geometry: Introduction and contentsDistance between two pointsMidpoint. Coordinates of midpointEquation of a lineEquation of a planeDistance from point to planeDistance between two planesDistance from a point to a line - 2-DimensionalDistance from a point to a line - 3-DimensionalAngle between two planesAngle between line and plane

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