Segment Addition and Distance

Segment addition is really a simple concept.

Basically, the length of a line segment is the sum of all of the smaller segments joined together to form it. Formally, this concept is called the segment addition or ruler postulate.

If I ask you to determine how far it is from House A to House C, you can easily see that you need to add the distance from House A to House B (5 miles) to the distance from House B to House C (11 miles).

So, the distance from House A to House C is:
distance House A to House C = 5 miles + 11 miles
distance House A to House C = 16 miles.

Segment Notation

Of course we get tired of saying (not to mention writing) “the distance between House A and House C.” Instead we use symbols.

When we name a line segment with endpoints A and C using symbols, we write AC .

But if we want to denote the length of the line segment, or the distance between points A and C, we simply write AC.

Suppose that I tell you that point B is between A and C. Note, that does NOT mean that B is exactly in the middle of points A and C, just somewhere between. Regardless of the position of B, as long as it is between A and C then we can refer to the distance between points A and C using symbols in this way:
AC = AB + BC

Remember, it doesn’t matter where point B lies along AC. As long as B is between A and C the above equation is correct.

A line between points A and B, between points B and C, and a line from point A through point B to point C

Now, if we include actual lengths, measurements, on our picture, we can determine the measure of AC using our same equation.

AC = AB + BC

A line from A to B to C. Segment A to B has length 7 feet and segment B to C has length 13

AC = 7 + 13 Add the length of the two segments together

AC = 20 AB + BC = 7 + 13 = 20

 

Postulate 1-1 The Ruler Postulate

The points on any line can be paired with real numbers so that given any two points P and Q on the line, P corresponds to zero, and Q corresponds to a positive number.

A line containing point P with value 0 and point Q with value 1

If R is between P and Q, then PR + RQ = PQ. 

A line containing point P, R, and Q, with point R in the middle

If PR + RQ = PQ, then R is between P and Q.

Definition of Betweeness

Segments can be defined by using the idea of betweenness of points.

In the figure, Point B is between A and C.

For B to be between A and C, all three points must be collinear and B must lie on AC.

A line containing points A, B and C with point B in the middle

Notes

AB , consists of points A and B and all points between A and B. The measure of AC, written AB, is the distance between A and B. Thus, the measure of a line segment is the same as the distance between its two endpoints.

A line containing points A and B. The points are shown to form a line segment between them

Notes

A number line with point A at -3.5, point B at -2, point C at 1, and point D at 3

The distance between two points on a number line is the absolute value of the difference of the coordinates.

BC = | B - C |

BC = -2 - 1 Use the coordinates of B and C

BC = 3

The distance between point B and point C is counted to be 3

The distance between two points on a number line can also be found by counting the distance between the two points.

3 whole steps from B to C

Remember the distance is always positive.

The same number line as before, with point A at -3.5, point B at -2, point C at 1, and point D at 3

In math we call this the absolute value. Find the distance for each by counting.  

AB = 1.5, BA = 1.5      

CB = 3, BC = 3

BD = 5, DB = 5

Does it matter if you start with A or B to find AB? No

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