# Distance and Midpoint of a Line

## Distance Between Two Points

Linear distances is useful in maps and vectors, such as calculating the direct distance from one town to another.

To find the distance from one point to another, simply use the following formula:

$d=\sqrt { { ({ x }_{ 2 }-{ x }_{ 1 }) }^{ 2 }+{ ({ y }_{ 2 }-{ y }_{ 1 }) }^{ 2 } }$

This is basically a Pythagoras' Theorem in a more fancy way. The components ${ x }_{ 2 }-{ x }_{ 1 }$ is the difference of horizontal difference (hence the x) between the two points. On the other hand, the components ${ y }_{ 2 }-{ y }_{ 1 }$ is the difference of vertical distance between the two points.

Use the following Geogebra Applet to check out distances between the two points.

### Example:

Find the distance between the points (3, 4) and (-5,-2).

$d=\sqrt { { (-5-3) }^{ 2 }+{ (-2-4) }^{ 2 } } \\ \\ d=\sqrt { { (-8 })^{ 2 }+{ (-6) }^{ 2 } } \\ \\ d=\sqrt { 64+36 } \\ \\ d=\sqrt { 100 } =\quad 10\quad units$

## Midpoint of Two Points

Use the following formula to find the midpoint of two points:

$Midpoint=\left( \cfrac { { x }_{ 1 }+{ x }_{ 2 } }{ 2 } ,\cfrac { { y }_{ 1 }+{ y }_{ 2 } }{ 2 } \right)$

The formula finds the halfway between the horizontal and vertical distances of the two points. Discover the midpoint using the Geogebra Applet below:

### Example:

Find the midpoint between (3, 4) and (-5, -2).

$Midpoint=\left( \cfrac { { x }_{ 1 }+{ x }_{ 2 } }{ 2 } ,\cfrac { { y }_{ 1 }+{ y }_{ 2 } }{ 2 } \right) \\ \\ Midpoint=\left( \cfrac { 3+-5 }{ 2 } ,\cfrac { 4+-2 }{ 2 } \right) \\ \\ Midpoint=\left( \cfrac { -2 }{ 2 } ,\cfrac { 2 }{ 2 } \right) \\ \\ Midpoint=\left( -1,1 \right)$