##### Finding the length of a straight line

### What is a straight line?

A straight line is the set of all points between and extending two points. In most geometries, a line is a primitive object that does not have formal properties beyond length, its single dimension.

The two properties of a straight line in Euclidian geometry:

- They have only one dimension
- One length
- They extend in two directions forever

Although lines are infinitely long, usually just a part of a line is considered. Any section of a line joining two points is called a line segment.

### What is a line segment?

A line segment is a segment, or finite portion of an infinite straight line.

If you know the co-ordinates of the end points of a line segment you can use Pythagoras theorem to calculate the length of the line segment.

### What is the length of a line segment?

The distance between two points is the length of the line segment connecting them. The distance between two points is always positive. Segments that have equal length are called congruent segments.

Example 1: Find the distance between the points (1,1) and (7,9).

$\displaystyle {{a}^{2}}={{b}^{2}}+{{c}^{2}}$ (__Pythagoras theorem__)

Work out each expression. Undo the square by taking the square root of both sides.

$\displaystyle {{a}^{2}}={{8}^{2}}+{{6}^{2}}$

$\displaystyle {{a}^{2}}=64+36$

$\displaystyle {{a}^{2}}=100$

$\displaystyle a=\sqrt{{100}}$

$\displaystyle a=10~units~$

Example 2: Given that A(3,6) and B(7,3), find the length of AB.

$\displaystyle A{{B}^{2}}=A{{C}^{2}}+C{{B}^{2}}$

$\displaystyle A{{B}^{2}}={{3}^{2}}+{{4}^{2}}$

$\displaystyle =9+16=25$

$\displaystyle AB=\sqrt{{25}}=5units$

$\displaystyle {{a}^{2}}={{b}^{2}}+{{c}^{2}}$ __(Pythagoras theorem) __Work out each expression.

### Midpoints

It is possible to find the co-ordinates of the midpoint of the line segment (the point that is exactly halfway between the two original points).

##### What is a midpoint?

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

Example 3: Consider the following line segment and the points A(3,4) and B(5,10).

If you add both x co-ordinates and then divide by two you get $\displaystyle \frac{{(3+5)}}{2}=\frac{8}{2}=4$

If you add both y co-ordinates and then divide by two you get $\displaystyle \frac{{(4+10)}}{2}=\frac{{14}}{2}=7$

This gives a new point with co-ordinates (4,7). This point is exactly half way between A and B.