1. 1.3 What you should learn Why you should learn it
Segments and Their Measures
What you should learn
GOAL 1
Use Segment Postulates
GOAL 2
Use the Distance Formula to measure distances
Why you should learn it
To solve real-life problems, such as finding distances along a diagonal city street.

2. 1.3 Segments and Their Measures 1 GOAL USING SEGMENT POSTULATES
In geometry, rules that are accepted without proof are called or
postulates
axioms
Rules that are proved are called
theorems

3. POSTULATES YOU NEED TO KNOW
RULER POSTULATE
The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the of the point.
coordinate
The between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is also called the of AB.
distance
length A B
names of points
coordinates
of points x1 x2
AB = x2 – x1
EXAMPLE 1

4. Extra Example 1
Measure the green bar on page 17 (it has the word postulate in it) to the nearest millimeter. Then measure it again, this time placing your ruler with the 2 at one end of the bar. If you understand the Ruler Postulate, you’ll get the same measurement as before.
Your answer should be
about 133 mm.

5. SEGMENT ADDITION POSTULATE If B is between A and C, then AB + BC = AC.
Also, if AB + BC = AC, then B is between A and C.
(Remember: “between” implies the points are collinear. AB BC A B C AC
Do you see that the length of the blue and red segments added together is equal to the length of the purple segment?
EXAMPLE 2

6. Extra Example 2
Two friends leave their homes and walk in a straight line toward the other’s home. When they meet one has walked 425 meters and the other has walked 267 meters. How far apart are their homes? Click for a hint.
425 m m
Answer:
The solution is 425 m m = 692 m.

7. Checkpoint
Measure the length of BC at the top of page 18 to the nearest millimeter.
A car with a trailer has a total length of 27 feet. If the trailer has a total length of 13 feet, how long is the car?
1. about 25 mm ft

8. 1.3 Segments and Their Measures 2 GOAL USING THE DISTANCE FORMULA
To find the distance between two points in a coordinate plane, we use the
Distance Formula

9. THE DISTANCE FORMULA
If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is y x
A(x1, y1)
B(x2, y2)
Important!!! Pay close attention to where the coordinates fit in the formula!
EXAMPLE 3

10. Extra Example 3
Find the lengths of the segments. Tell whether any of the segments have the same length. Click for each answer. y 1 x
E(-3, 3)
G(-3, 0)
F(1, 2)
H(0, -1)
None of the segments have the same length.

11. Checkpoint Find the distance between each pair of points. y K(-2, 2) 1x
K(-2, 2)
L(-2, -3)
M(0, -1) y
Click for each answer.

12. Be sure you understand this concept!
Of course, some segments have equal lengths. These are called _________________.
congruent segments
Important: Segments are NOT equal; they are congruent.
Congruent segments have equal lengths. M N P Q
Incorrect:
Correct:
Be sure you understand this concept!

13. Now let’s look again at the Distance Formula.y x
B(x1, y1)
A(x2, y2)
Click to form a right triangle. a b c C
What are the coordinates of C?
(x2, y1)
Square both sides of the distance formula:
Let’s say AB= c, BC = a, and AC = b.
From the Ruler Postulate, we also know that BC = x2 – x1 and AC = y2 –y1 . Then by substitution we know that c2 = a2 + b2. This is known as the Pythagorean Theorem. More in Chapter 9!

14. Study Example 4 before going on!
Extra Example 4
C(0, 740)
D(2050, -370) y x
370
- 410
On the map, the city blocks are 410 feet apart east-west and 370 feet apart north-south.
Find the walking distance between C and D.
Solution:
What would the distance be if a diagonal street existed between the two points?
Solution:

15. Checkpoint y x E(820, 0) F(-410, -1110)
370
- 410
Find the diagonal distance between points E and F on the map.
Answer:
about 1657 ft

16. QUESTIONS?