1.2: Segments and Distance
Learning Objectives
 Use the ruler postulate.
 Use the segment addition postulate.
 Plot line segments on the
x−y plane.
Review Queue
 Draw a line segment with endpoints
C andD .  How would you label the following figure? List 2 different ways.
 Draw three collinear points and a fourth that is coplanar.
 Plot the following points on the
x−y plane. (3, 3)
 (4, 2)
 (0, 7)
 (6, 0)
Know What? The average adult human body can be measured in “heads.” For example, the average human is 78 heads tall. When doing this, each person uses their own head to measure their own body. Other measurements are in the picture to the right.
See if you can find the following measurements:
 The length from the wrist to the elbow
 The length from the top of the neck to the hip
 The width of each shoulder
Measuring Distances
Distance: The length between two points.
Measure: To determine how far apart two geometric objects are.
The most common way to measure distance is with a ruler. In this text we will use inches and centimeters.
Example 1: Determine how long the line segment is, in inches. Round to the nearest quarterinch.
Solution: To measure this line segment, it is very important to line up the “0” with the one of the endpoints. DO NOT USE THE EDGE OF THE RULER.
From this ruler, it looks like the segment is 4.75 inches (in) long.
Inchrulers are usually divided up by eightinch (or 0.125 in) segments. Centimeter rulers are divided up by tenthcentimeter (or 0.1 cm) segments.
The two rulers above are NOT DRAWN TO SCALE, which means that the measured length is not the distance apart that it is labeled.
Example 2: Determine the measurement between the two points to the nearest tenth of a centimeter.
Solution: Even though there is no line segment between the two points, we can still measure the distance using a ruler.
It looks like the two points are 6 centimeters (cm) apart.
NOTE: We label a line segment,
Label It  Say It 


The distance between 

The measure of 
Ruler Postulate
Ruler Postulate: The distance between two points is the absolute value of the difference between the numbers shown on the ruler.
The ruler postulate implies that you do not need to start measuring at “0”, as long as you subtract the first number from the second. “Absolute value” is used because distance is always positive.
Example 3: What is the distance marked on the ruler below? The ruler is in centimeters.
Solution: 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 4: Draw
Solution: To draw a line segment, start at “0” and draw a segment to 3.825 in.
Put points at each end and label.
Segment Addition Postulate
First, in the picture below,
Segment Addition Postulate: If
For example, if
Example 5: Make a sketch of
Solution: Draw
Example 6: In the picture from Example 5, if
Solution: Use the Segment Additional Postulate.
Example 7: Make a sketch of:
Solution: Interpret the first sentence first:
Then add in what we know about
Example 8: Find
Solution:
For SV: It is equal to
Example 9: Algebra Connection For
Solution: Use the Segment Addition Postulate.
Distances on a Grid
You can now find the distances between points in the
If the line is vertical, find the change in the
If the line is horizontal, find the change in the
Example 10: What is the distance between the two points shown below?
Solution: Because this line is vertical, look at the change in the
The distance between the two points is 6 units.
Example 11: What is the distance between the two points shown below?
Solution: Because this line is horizontal, look at the change in the
The distance between the two points is 7 units.
Know What? Revisited The length from the wrist to the elbow is one head, the length from the top of the neck to the hip is two heads, and the width of each shoulder one head width.
Review Questions
 Questions 18 are similar to Examples 1 and 2.
 Questions 912 are similar to Example 3.
 Questions 1317 are similar to Examples 5 and 6.
 Questions 18 and 19 are similar to Example 7 and 8.
 Questions 20 and 21 are similar to Example 9.
 Questions 2226 are similar to Examples 10 and 11.
For 14, find the length of each line segment in inches. Round to the nearest
For 58, find the distance between each pair of points in centimeters. Round to the nearest tenth.
For 912, use the ruler in each picture to determine the length of the line segment.
 Make a sketch of
BT¯¯¯¯¯¯¯ , withA betweenB andT .  If
O is in the middle ofLT¯¯¯¯¯¯¯ , where exactly is it located? IfLT=16 cm , what isLO andOT ?  For three collinear points,
A betweenT andQ . Draw a sketch.
 Write the Segment Addition Postulate.
 If
AT=10 in andAQ=5 in , what isTQ ?
 For three collinear points,
M betweenH andA . Draw a sketch.
 Write the Segment Addition Postulate.
 If
HM=18 cm andHA=29 cm , what isAM ?
 For three collinear points,
I betweenM andT . Draw a sketch.
 Write the Segment Addition Postulate.
 If
IT=6 cm andMT=25 cm , what isAM ?
 Make a sketch that matches the description:
B is betweenA andD .C is betweenB andD .AB=7 cm, AC=15 cm , andAD=32 cm . FindBC,BD , andCD .  Make a sketch that matches the description:
E is betweenF andG .H is betweenF andE .FH=4 in, EG=9 in , andFH=HE . FindFE,HG , andFG .
For 20 and 21, Suppose \begin{align*}J\end{align*} is between \begin{align*}H\end{align*} and \begin{align*}K\end{align*}. Use the Segment Addition Postulate to solve for \begin{align*}x\end{align*}. Then find the length of each segment.
 \begin{align*}HJ = 4x + 9, \ JK = 3x + 3, \ KH = 33\end{align*}
 \begin{align*}HJ = 5x  3, \ JK = 8x  9, \ KH = 131\end{align*}
For 2326, determine the vertical or horizontal distance between the two points.
Review Queue Answers
 line \begin{align*}l, \ \overline{MN}\end{align*}