What if you were given the coordinates of two points that form either a vertical or horizontal line? How would you determine how far apart those two points are? After completing this Concept, you'll be able to determine the distance between two such points.
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James Sousa: Ruler Postulate and Segment Addition Postulate
Guidance
Distance is the measure of length between two points. To measure is to determine how far apart two geometric objects are. The most common way to measure distance is with a ruler. Inchrulers are usually divided up by eighthinch (or 0.125 in) segments. Centimeter rulers are divided up by tenthcentimeter (or 0.1 cm) segments. Note that the distance between two points is the absolute value of the difference between the numbers shown on the ruler. This implies that you do not need to start measuring at “0”, as long as you subtract the first number from the second.
The segment addition postulate states that if , , and are collinear and is between and , then .
You can find the distances between points in the – plane if the lines are horizontal or vertical. If the line is vertical, find the change in the coordinates. If the line is horizontal, find the change in the coordinates.
Example A
What is the distance marked on the ruler below? The ruler is in centimeters.
Subtract one endpoint from the other. The line segment spans from 3 cm to 8 cm.
The line segment is 5 cm long. Notice that you also could have done .
Example B
Make a sketch of , where is between and .
Draw first, then place on the segment.
Example C
What is the distance between the two points shown below?
Because this line is vertical, look at the change in the coordinates.
The distance between the two points is 6 units.
Guided Practice
1. Draw , such that .
2. In the picture from Example B, if and , what is ?
3. What is the distance between the two points shown below?
Answers:
1. To draw a line segment, start at “0” and draw a segment to 3.825 in.
Put points at each end and label.
2. Use the Segment Addition Postulate.
3. Because this line is horizontal, look at the change in the coordinates.
The distance between the two points is 7 units.
Practice
For 14, use the ruler in each picture to determine the length of the line segment.
 Make a sketch of , with between and .
 If is in the middle of , where exactly is it located? If , what is and ?

For three collinear points,
between
and
:
 Draw a sketch.
 Write the Segment Addition Postulate for your sketch.
 If and , what is ?

For three collinear points,
between
and
:
 Draw a sketch.
 Write the Segment Addition Postulate for your sketch.
 If and , what is ?

For three collinear points,
between
and
:
 Draw a sketch.
 Write the Segment Addition Postulate for your sketch.
 If and , what is ?
 Make a sketch that matches the description: is between and . is between and . , and . Find , and .
 Make a sketch that matches the description: is between and . is between and . , and . Find , and .
For 12 and 13, Suppose is between and . Use the Segment Addition Postulate to solve for . Then find the length of each segment.
For 1417, determine the vertical or horizontal distance between the two points.
 Make a sketch of: is between and . is between and . , and .
 Find and from #18.
 For , suppose that is between and . If , and , find .