1.04 Segments

	Blueprints for this Lesson:
	Find lengths of segments. Solve problems involving segment measures.

Foundational Knowledge:

Try these on your own paper, and check your work with the answer link below.

Simplify the following:

|-3|
|-2 + 5|
|-4 - 7|

Solve for x:
2x + 8 = -14
x - 5 = 3x - 13

Click here to check your work.

Framework For Understanding:

Can you imagine drawing up a set of blueprints for a building and not using at least one segment in the drawing? It would be almost impossible. Everything would have to be made with arcs or circles. Think about how many geometric shapes are created using segments. So, whether you are working on a building or on a geometry problem, it is very important that you know everything you can about segments.

Finding the Lengths of Segments:

A numberline is drawn with point A at 3 and point B at 6. Under the numberline a ruler has been placed so that point A is at the 3 inch mark and point B is at the 6 inch mark.

Find the distance between points A and B on the number line above. This is a lot like finding the length of something using a ruler.

length of segment A B equals the absolute value of 6 minus 3

Did you notice the absolute value signs written around the | 6 - 3 |? These signs allow us to write the subtraction problem as 6 - 3 or 3 - 6. Either way, the absolute values will be the same and the answer will be the distance between the two points.

Postulate 7
On a ruler or number line, you can find the distance between two points by subtracting their coordinates and then taking the absolute value.

Did you know that two segments with the same length are called congruent equals segment ?

If the lengths of line segment A B and line segment C D are equal, then and must be congruent. We could write this using only geometry symbols.

If AB = CD, then segment A B is congruent to segment C D .

Because AB = CD, we can mark the segments in the same way to illustrate that they are congruent. Roll your cursor over the image to the right to see the segments marked to show congruence.

segment A B is drawn and labeled as 3 cm, segment C D is drawn and labeled 3 cm, the two segments are therefore congruent.

Segment Addition:

If three points are on the same line, then one point is between the other two. This is the definition of betweeness. This leads us to another postulate.

Postulate 8 - Segment Addition Postulate
If point C is between points A and B, then AC + CB = AB.

Line segment A B drawn with point C between A and B on the segment.

This postulate is helpful when we are trying to find missing information about lengths of segments.

Example: Q is between P and R with PQ = 2x - 1, QR = 3x + 5, and PR = 34. Write an equation to solve for x, then find PQ and QR. Make sure to check your solutions.

Line segment P R is drawn with point Q between points P and R on the segment. Segment P Q is labeled 2 times x minus 1 while segment Q R is labeled 3 times x plus 5.

Before we can start solving this problem, we need to look at the illustration and identify some sort of equation. Because of the segment addition postulate, we know that PQ + QR = PR. Now that we have our equation, let's solve it.

PQ + QR = PR (2x - 1) + (3x + 5) = 34

Step 1: Replace PQ, QR and PR with the given information.

PQ = 2x - 1
QR = 3x + 5
PR = 34

2x - 1 + 3x + 5 =
5x + 4 =
5x =
x =

34
34
30
6

Step 2: Solve the equation for x.

PQ	=	2x - 1
	=	2(6) - 1
	=	12 - 1
	=	11
PQ	=	11

QR	=	3x + 5
	=	3(6) + 5
	=	18 + 5
	=	23
QR	=	23

Step 3: Substitute the value of x into the expressions for PQ and QR and simplify.

Solution: x = 6, PQ = 11 and QR = 23

Step 4: Answer the question.

Check:

PQ + QR = PR

11 + 23 = 34

Step 5: Check your solution. Substitute the values of PQ and QR into the original equation to verify your answer.

Midpoint:

What do you think the term "midpoint" means? The word is made up of two parts, "mid" and "point."

The midpoint of a segment is the point that divides the segment into two equal parts. The midpoint represents the middlemost point between two endpoints. Knowing this definition, what type of equation can we set up to solve the following problem?

Example: Y is the midpoint of Line X Z . Find XY, YZ and XZ.

Line segment X Z is drawn so that Y is between X and Z on the segment. Segment X Y is labeled 5 times n plus 1. Segment Y Z is labeled 3 times n plus 7.

Because Y is given as the midpoint of segment X Z , we know that XY must equal YZ. In other words XY = YZ. Let's plug into our equation what we know about XY and YZ and solve for n. Once we know the value of n, we can find the measurements for XY, YZ and XZ.

XY	=	YZ
(5n + 1)	=	(3n + 7)

Step 1: Replace XY and YZ with the given information.

XY = 5n + 1
YZ = 3n + 7

5n + 1	=	3n + 7
2n + 1	=	7
2n	=	6
n	=	3

Step 2: Solve the equation for n.

XY	=	5n + 1
	=	5(3) + 1
	=	15 + 1
	=	16

YZ	=	3n + 7
	=	3(3) + 7
	=	9 + 7
	=	16

Step 3: Substitute the value of n into the expressions for XY and YZ and simplify.

Solution:
XY = 16, YZ = 16, XZ = 32

Step 4: Answer the question.

Check: XY = YZ
16 = 16

and XY + YZ = XZ
16 + 16 = 32

Step 5: Check your solution. Substitute the values of PQ and QR into the original equation to verify your answer.

Now let's discuss other ways to cut a segment into two congruent parts.

Segment Bisector:

A segment bisector is a segment, ray, line, or plane that intersects a segment at its midpoint.

Plane Q is drawn so that segment E F, line N O and ray S R intersect the plane and each other only at point H on the plane. Segment I H is on the plane.

It is time to move on and practice what you have learned.