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.

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

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

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.

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

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 267 m Answer: The solution is 425 m + 267 m = 692 m.

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 2. 14 ft

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

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

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.

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

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!

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!

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:

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

QUESTIONS?