Geometry (A): READ: Segment and Rays Reading

Segments and Distance

Learning Objectives

Measure distances using different tools.
Understand and apply the ruler postulate to measurement.
Understand and apply the segment addition postulate to measurement.
Use endpoints to identify distances on a coordinate grid.

Introduction

You have been using measurement for most of your life to understand quantities like weight, time, distance, area, and volume. Any time you have cooked a meal, bought something, or played a sport, measurement has played an important role. This lesson explores the postulates about measurement in geometry.

Measuring Distances

There are many different ways to identify measurements. This lesson will present some that may be familiar, and probably a few that are new to you. Before we begin to examine distances, however, it is important to identify the meaning of distance in the context of geometry. The distance between two points is defined by the length of the line segment that connects them.

The most common way to measure distance is with a ruler. Also, distance can be estimated using scale on a map. Practice this skill in the example below.

Notation Notes: When we name a segment we use the endpoints and and overbar with no arrows. For example, "Segment $AB$ " is written $\overline{AB}$ . The length of a segment is named by giving the endpoints without using an overline. For example, the length of $\overline{AB}$ is written $AB$ . In some books you may also see $m \overline{AB}$ , which means the same as $AB$ , that is, it is the length of the segment with endpoints $A$ and $B$ .

Example 1

Use the scale to estimate the distance between Aaron’s house and Bijal’s house. Assume that the first third of the scale in black represents one inch.

You need to find the distance between the two houses in the map. The scale shows a sample distance. Use the scale to estimate the distance. You will find that approximately three segments the length of the scale fit between the two points. Be careful—three is not the answer to this problem! As the scale shows one unit equal to two miles, you must multiply three units by two miles.

$\mbox{3 units} \times \frac{\mbox{2 miles}} {\mbox{1 unit}} = \mbox{6 miles}$

The distance between the houses is about six miles.

You can also use estimation to identify measurements in other geometric figures. Remember to include words like approximately, about, or estimation whenever you are finding an estimated answer.

Ruler Postulate

You have probably been using rulers to measure distances for a long time and you know that a ruler is a tool with measurement markings.

Ruler Postulate: If you use a ruler to find the distance between two points, the distance will be 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 the zero mark, as long as you use subtraction to find the distance. Note, we say “absolute value” here since distances in geometry must always be positive, and subtraction can yield a negative result.

Example 2

What distance is marked on the ruler in the diagram below? Assume that the scale is marked in centimeters.

The way to use the ruler is to find the absolute value of difference between the numbers shown. The line segment spans from $3 \;\mathrm{cm}$ to $8 \;\mathrm{cm}$ .

$|3 - 8| = |-5| = 5$

The absolute value of the difference between the two numbers shown on the ruler is $5 \;\mathrm{cm}$ . So, the line segment is $5 \;\mathrm{cm}$ long.

Example 3

Use a ruler to find the length of the line segment below.

Line up the endpoints with numbers on your ruler and find the absolute value of the difference between those numbers. If you measure correctly, you will find that this segment measures $2.5\;\mathrm{inches}$ or $6.35\;\mathrm{centimeters}$ .

Segment Addition Postulate

Segment Addition Postulate: The measure of any line segment can be found by adding the measures of the smaller segments that comprise it.

That may seem like a lot of confusing words, but the logic is quite simple. In the diagram below, if you add the lengths of $\overline{AB}$ and $\overline{BC}$ , you will have found the length of $\overline{AC}$ . In symbols, $AB+BC=AC$ .

Use the segment addition postulate to put distances together.

Example 4

The map below shows the distances between three collinear towns. Assume that the first third of the scale in black represents one inch.

What is the distance between town $1$ and town $3$ ?

You can see that the distance between town $1$ and town $2$ is eight miles. You can also see that the distance between town $2$ and town $3$ is five miles. Using the segment addition postulate, you can add these values together to find the total distance between town $1$ and town $3$ .

$8+5=13$

The total distance between town $1$ and town $3$ is $13\;\mathrm{miles}$ .

Congruent Line Segments

One of the most important words in geometry is congruent. This term refers to geometric objects that have exactly the same size and shape. Two segments are congruent if they have the same length.

Notation Notes:

When two things are congruent we use the symbol $\cong$ . For example if $\overline{AB}$ is congruent to $\overline{CD}$ , then we would write $\overline{AB} \cong \overline{CD}$ .
When we draw congruent segments, we use tic marks to show that two segments are congruent.
If there are multiple pairs of congruent segments (which are not congruent to each other) in the same picture, use two tic marks for the second set of congruent segments, three for the third set, and so on. See the two following illustrations.

Recall that the length of segment $\overline{AB}$ can be written in two ways: $m\overline{AB}$ or simply $AB$ . This might be a little confusing at first, but it will make sense as you use this notation more and more. Let’s say we used a ruler and measured $\overline{AB}$ and we saw that it had a length of $5 \;\mathrm{cm}$ . Then we could write $m\overline{AB} = 5 \;\mathrm{cm}$ , or $AB = 5 \;\mathrm{cm}$ .

If we know that $\overline{AB} \cong \overline{CD}$ , then we can write $m\overline{AB} = m\overline{CD}$ or simply $AB = CD$ .

You can prove two segments are congruent in a number of ways. You can measure them to find their lengths using any units of measurement—the units do not matter as long as you use the same units for both measurements. Or, if the segments are drawn in the $x-y$ plane, you can also find their lengths on the coordinate grid. Later in the course you will learn other ways to prove two segments are congruent.

Example 1

Henrietta drew a line segment on a coordinate grid as shown below.

She wants to draw another segment congruent to the first that begins at $(-1,1)$ and travels straight up (that is, in the $+y$ direction). What will be the coordinates of its second endpoint?

You will have to solve this problem in stages. The first step is to identify the length of the segment drawn onto the grid. It begins at $(2,3)$ and ends at $(6,3)$ . So, its length is $4\;\mathrm{units}$ .

The next step is to draw the second segment. Use a pencil to create the segment according to the specifications in the problem. You know that the segment needs to be congruent to the first, so it will be $4\;\mathrm{units}$ long. The problem also states that it travels straight up from the point $(-1,1)$ . Draw in the point at $(-1,1)$ and make a line segment $4\;\mathrm{units}$ long that travels straight up.

Now that you have drawn in the new segment, use the grid to identify the new endpoint. It has an $x-$ coordinate of $-1$ and a $y-$ coordinate of $5$ . So, its coordinates are $(-1,5)$ .

Segment Midpoints

Now that you understand congruent segments, there are a number of new terms and types of figures you can explore. A segment midpoint is a point on a line segment that divides the segment into two congruent segments. So, each segment between the midpoint and an endpoint will have the same length. In the diagram below, point $B$ is the midpoint of segment $\overline{AC}$ since $\overline{AB}$ is congruent to $\overline{BC}$ .

There is even a special postulate dedicated to midpoints.

Segment Midpoint Postulate: Any line segment will have exactly one midpoint—no more, and no less.

Example 2

Nandi and Arshad measure and find that their houses are $10\;\mathrm{miles}$ apart. If they agree to meet at the midpoint between their two houses, how far will each of them travel?

The easiest way to find the distance to the midpoint of the imagined segment connecting their houses is to divide the length by $2$ .

$10 \div 2 = 5$

So, each person will travel five miles to meet at the midpoint between Nandi’s and Arshad’s houses.

Segment Bisectors

Now that you know how to find midpoints of line segments, you can explore segment bisectors. A bisector is a line, segment, or ray that passes through a midpoint of another segment. You probably know that the prefix “bi” means two (think about the two wheels of a bicycle). So, a bisector cuts a line segment into two congruent parts.

Example 3

Use a ruler to draw a bisector of the segment below.

The first step in identifying a bisector is finding the midpoint. Measure the line segment to find that it is $4 \;\mathrm{cm}$ long. To find the midpoint, divide this distance by $2$ .

$4 \div 2 = 2$

So, the midpoint will be $2 \;\mathrm{cm}$ from either endpoint on the segment. Measure $2 \;\mathrm{cm}$ from an endpoint and draw the midpoint.

To complete the problem, draw a line segment that passes through the midpoint. It doesn’t matter what angle this segment travels on. As long as it passes through the midpoint, it is a bisector.

Lesson Summary

In this lesson, we explored segments and distances. Specifically, we have learned:

How to measure distances using many different tools.
To understand and apply the Ruler Postulate to measurement.
To understand and apply the Segment Addition Postulate to measurement.

These skills are useful whenever performing measurements or calculations in diagrams. Make sure that you fully understand all concepts presented here before continuing in your study.

The following questions are for your own benefit. The answers are listed below in order for you to check your work.

Review Questions

Use a ruler to measure the length of $\overline{AB}$ below.
According to the ruler in the following image, how long is the cockroach?
The ruler postulate says that we could have measured the cockroach in $2$ without using the $0 \;\mathrm{cm}$ marker as the starting point. If the same cockroach as the one in $2$ had its head at $6.5 \;\mathrm{cm}$ , where would its tail be on the ruler?
Suppose $M$ is exactly in the middle of $\overline{PQ}$ and $PM = 8 \;\mathrm{cm}$ . What is $PQ?$
What is $CE$ in the diagram below?
Find $x$ in the diagram below:
True or false: If $AB = 5\;\mathrm{cm}$ and $BC = 12\;\mathrm{cm}$ , then $AC = 17\;\mathrm{cm}$ .
True or false: $|a-b| = |b-a|$ .
One of the statements in 8 or 9 is false. Show why it is false, and then change the statement to make it true.

Review Answers

Answers will vary depending on scaling when printed and the units you use.
$4.5 \;\mathrm{cm}$ (yuck!).
The tail would be at either $11 \;\mathrm{cm}$ or $2 \;\mathrm{cm}$ , depending on which way the cockroach was facing.
$PQ = 2 \;\mathrm{(PM)} = 16 \;\mathrm{cm}$ .
$CE = 3 \;\mathrm{ft} + 9 \;\mathrm{ft} = 12 \;\mathrm{ft}$ .
$x = 36 \;\mathrm{km} - 7 \;\mathrm{km} = 29 \;\mathrm{km}$ .
False.
True. $a - b = - (b - a)$ , but the absolute value sign makes them both positive.
Number 8 is false. See the diagram below for a counterexample. To make 8 true, we need to add something like: “If points $A$ , $B$ , and $C$ are collinear, and $B$ is between $A$ and $C$ , then if $AB = 5 \;\mathrm{cm}$ and $BC = 12 \;\mathrm{cm}$ , then $AC = 17 \;\mathrm{cm}$ .”

Last modified: Monday, June 28, 2010, 11:40 AM